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A Mathematical Introduction to Logic Second Edition Herbert B. Enderton University of California, Los Angeles

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Contents PREFACE

ix

INTRODUCTION

xi

CHAPTER ZERO

Useful Facts about Sets

CHAPTER ONE 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7

Sentential Logic Informal Remarks on Formal Languages The Language of Sentential Logic Truth Assignments A Parsing Algorithm Induction and Recursion Sentential Connectives Switching Circuits Compactness and Effectiveness

CHAPTER TWO 2.0 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8

First-Order Logic Preliminary Remarks First-Order Languages Truth and Models A Parsing Algorithm A Deductive Calculus Soundness and Completeness Theorems Models of Theories Interpretations Between Theories Nonstandard Analysis

CHAPTER THREE 3.0 3.1 3.2 3.3 3.4

Undecidability Number Theory Natural Numbers with Successor Other Reducts of Number Theory A Subtheory of Number Theory Arithmetization of Syntax

1 11 11 13 20 29 34 45 54 59 67 67 69 80 105 109 131 147 164 173 182 182 187 193 202 224 vii

viii 3.5 3.6 3.7 3.8

Contents Incompleteness and Undecidability Recursive Functions Second Incompleteness Theorem Representing Exponentiation

234 247 266 276

CHAPTER FOUR Second-Order Logic 4.1 Second-Order Languages 4.2 Skolem Functions 4.3 Many-Sorted Logic 4.4 General Structures

282 282 287 295 299

SUGGESTIONS FOR FURTHER READING

307

LIST OF SYMBOLS

309

INDEX

311

Preface

T

his book, like the ﬁrst edition, presents the basic concepts and results of logic: the topics are proofs, truth, and computability. As before, the presentation is directed toward the reader with some mathematical background and interests. In this revised edition, in addition to numerous “local” changes, there are three “global” ways in which the presentation has been changed: First, I have attempted to make the material more accessible to the typical undergraduate student. In the main development, I have tried not to take for granted information or insights that might be unavailable to a junior-level mathematics student. Second, for the instructor who wants to ﬁt the book to his or her course, the organization has been made more ﬂexible. Footnotes at the beginning of many of the sections indicate optional paths the instructor — or the independent reader — might choose to take. Third, theoretical computer science has inﬂuenced logic in recent years, and some of that inﬂuence is reﬂected in this edition. Issues of computability are taken more seriously. Some material on ﬁnite models has been incorporated into the text. The book is intended to serve as a textbook for an introductory mathematics course in logic at the junior-senior level. The objectives are to present the important concepts and theorems of logic and to explain their signiﬁcance and their relationship to the reader’s other mathematical work. As a text, the book can be used in courses anywhere from a quarter to a year in length. In one quarter, I generally reach the material on models of ﬁrst-order theories (Section 2.6). The extra time afforded by a semester would permit some glimpse of undecidability, as in Section 3.0. In a second ix

x

Preface

term, the material of Chapter 3 (on undecidability) can be more adequately covered. The book is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. There are no speciﬁc prerequisites aside from a willingness to function at a certain level of abstraction and rigor. There is the inevitable use of basic set theory. Chapter 0 gives a concise summary of the set theory used. One should not begin the book by studying this chapter; it is instead intended for reference if and when the need arises. The instructor can adjust the amount of set theory employed; for example it is possible to avoid cardinal numbers completely (at the cost of losing some theorems). The book contains some examples drawn from abstract algebra. But they are just examples, and are not essential to the exposition. The later chapters (Chapter 3 and 4) tend to be more demanding of the reader than are the earlier chapters. Induction and recursion are given a more extensive discussion (in Section 1.4) than has been customary. I prefer to give an informal account of these subjects in lectures and have a precise version in the book rather than to have the situation reversed. Exercises are given at the end of nearly all the sections. If the exercise bears a boldface numeral, then the results of that exercise are used in the exposition in the text. Unusually challenging exercises are marked with an asterisk. I cheerfully acknowledge my debt to my teachers, a category in which I include also those who have been my colleagues or students. I would be pleased to receive comments and corrections from the users of this book. The Web site for the book can be found at http://www.math.ucla.edu/∼ hbe/amil.

Introduction

S

ymbolic logic is a mathematical model of deductive thought. Or at least that was true originally; as with other branches of mathematics it has grown beyond the circumstances of its birth. Symbolic logic is a model in much the same way that modern probability theory is a model for situations involving chance and uncertainty. How are models constructed? You begin with a real-life object, for example an airplane. Then you select some features of this original object to be represented in the model, for example its shape, and others to be ignored, for example its size. And then you build an object that is like the original in some ways (which you call essential) and unlike it in others (which you call irrelevant). Whether or not the resulting model meets its intended purpose will depend largely on the selection of the properties of the original object to be represented in the model. Logic is more abstract than airplanes. The real-life objects are certain “logically correct” deductions. For example, All men are mortal. Socrates is a man. Therefore, Socrates is mortal. The validity of inferring the third sentence (the conclusion) from the ﬁrst two (the assumptions) does not depend on special idiosyncrasies of Socrates. The inference is justiﬁed by the form of the sentences rather than by empirical facts about mortality. It is not really important here what “mortal” means; it does matter what “all” means. Borogoves are mimsy whenever it is brillig. It is now brillig, and this thing is a borogove. Hence this thing is mimsy. xi

xii

Introduction

Again we can recognize that the third sentence follows from the ﬁrst two, even without the slightest idea of what a mimsy borogove might look like. Logically correct deductions are of more interest than the above frivolous examples might suggest. In fact, axiomatic mathematics consists of many such deductions laid end to end. These deductions made by the working mathematician constitute real-life originals whose features are to be mirrored in our model. The logical correctness of these deductions is due to their form but is independent of their content. This criterion is vague, but it is just this sort of vagueness that prompts us to turn to mathematical models. A major goal will be to give, within a model, a precise version of this criterion. The questions (about our model) we will initially be most concerned with are these: 1. What does it mean for one sentence to “follow logically” from certain others? 2. If a sentence does follow logically from certain others, what methods of proof might be necessary to establish this fact? 3. Is there a gap between what we can prove in an axiomatic system (say for the natural numbers) and what is true about the natural numbers? 4. What is the connection between logic and computability? Actually we will present two models. The ﬁrst (sentential logic) will be very simple and will be woefully inadequate for interesting deductions. Its inadequacy stems from the fact that it preserves only some crude properties of real-life deductions. The second model (ﬁrst-order logic) is admirably suited to deductions encountered in mathematics. When a working mathematician asserts that a particular sentence follows from the axioms of set theory, he or she means that this deduction can be translated to one in our model. This emphasis on mathematics has guided the choice of topics to include. This book does not venture into many-valued logic, modal logic, or intuitionistic logic, which represent different selections of properties of real-life deductions. Thus far we have avoided giving away much information about what our model, ﬁrst-order logic, is like. As brief hints, we now give some examples of the expressiveness of its formal language. First, take the English sentence that asserts the set-theoretic principle of extensionality, “If the same things are members of a ﬁrst object as are members of a second object, then those objects are the same.” This can be translated into our ﬁrst-order language as ∀ x ∀ y( ∀ z(z ∈ x ↔ z ∈ y) → x = y). As a second example, we can translate the sentence familiar to calculus students, “For every positive number ε there is a positive number δ such that for any x whose distance from a is less than δ, the distance between f (x) and b is less than ε” as ∀ ε(ε > 0 → ∃ δ(δ > 0 ∧ ∀ x(d xa < δ → d f xb < ε))). We have given some hints as to what we intend to do in this book. We should also correct some possible misimpressions by saying what we are not going to do. This book does not propose to teach the reader how to think. The word “logic” is sometimes used to refer to remedial thinking, but not by us. The reader already knows how to think. Here are some intriguing concepts to think about.

Chapter

Z E R O

Useful Facts about Sets

W

e assume that the reader already has some familiarity with normal everyday set-theoretic apparatus. Nonetheless, we give here a brief summary of facts from set theory we will need; this will at least serve to establish the notation. It is suggested that the reader, instead of poring over this chapter at the outset, simply refer to it if and when issues of a set-theoretic nature arise in later chapters. The author’s favorite book on set theory is of course his Elements of Set Theory (see the list of references at the end of this book). First a word about jargon. Throughout the book we will utilize an assortment of standard mathematical abbreviations. We use “” to signify the end of a proof. A sentence “If . . . , then . . .” will sometimes be abbreviated “. . . ⇒ . . . .” We also have “⇐” for the converse implication (for the peculiar way the word “implication” is used in mathematics). For “if and only if” we use the shorter “iff” (this has become part of the mathematical language) and the symbol “⇔.” For the word “therefore” we have the “∴” abbreviation. The notational device that extracts “x =

y” as the denial of “x = y” and “x ∈ / y” as the denial of “x ∈ y” will be extended to other cases. For example, in Section 1.2 we deﬁne “ |= τ ”; then “ |= τ ” is its denial. Now then, a set is a collection of things, called its members or elements. As usual, we write “t ∈ A” to say that t is a member of A, and “t ∈ / A” to say that t is not a member of A. We write “x = y” to 1

2

A Mathematical Introduction to Logic mean that x and y are the same object. That is, the expression “x” on the left of the equals sign is a name for the same object as is named by the other expression “y.” If A = B, then for any object t it is automatically true that t ∈ A iff t ∈ B. This holds simply because A and B are the same thing. The converse is the principle of extensionality: If A and B are sets such that for every object t, t∈A

iff

t ∈ B,

then A = B. This reﬂects the idea of what a set is; a set is determined just by its members. A useful operation is that of adjoining one extra object to a set. For a set A, let A; t be the set whose members are (i) the members of A, plus (ii) the (possibly new) member t. Here t may or may not already belong to A, and we have A; t = A ∪ {t} using notation deﬁned later, and t∈A

iff

A; t = A.

One special set is the empty set ∅, which has no members at all. Any other set is said to be nonempty. For any object x there is the singleton set {x} whose only member is x. More generally, for any ﬁnite number x1 , . . . , xn of objects there is the set {x1 , . . . , xn } whose members are exactly those objects. Observe that {x, y} = {y, x}, as both sets have exactly the same members. We have only used different expressions to denote the set. If order matters, we can use ordered pairs (discussed later). This notation will be stretched to cover some simple inﬁnite cases. For example, {0, 1, 2, . . .} is the set N of natural numbers, and {. . . , −2, −1, 0, 1, 2, . . .} is the set Z of all integers. We write “{x | x }” for the set of all objects x such that x . We will take considerable liberty with this notation. For example, { m, n | m < n in N} is the set of all ordered pairs of natural numbers for which the ﬁrst component is smaller than the second. And {x ∈ A | x } is the set of all elements x in A such that x . If A is a set all of whose members are also members of B, then A is a subset of B, abbreviated “A ⊆ B.” Note that any set is a subset of itself. Also, ∅ is a subset of every set. (“∅ ⊆ A” is “vacuously true,” since the task of verifying, for every member of ∅, that it also belongs to A requires doing nothing at all. Or from another point of view, “A ⊆ B” can be false only if some member of A fails to belong to B. If A = ∅, this is impossible.) From the set A we can form a new set, the power set P A of A, whose members are the subsets of A. Thus P A = {x | x ⊆ A}.

Chapter 0:

Useful Facts about Sets

3

For example, P∅ = {∅}, P{∅} = {∅, {∅}}. The union of A and B, A ∪ B, is the set of all things that are members of A or B (or both). For example, A; t = A ∪ {t}. Similarly, the intersection of A and B, A ∩ B, is the set of all things that are members of both A and B. Sets A and B are disjoint iff their intersection is empty (i.e., if they have no members in common). A collection of sets is pairwise disjoint iff any two members of the collection are disjoint. More generally, consider a set A whose members are themselves sets. The union, A, of A is the set obtained by dumping all the members of A into a single set: A = {x | x belongs to some member of A}. Similarly for nonempty A, A = {x | x belongs to all members of A}. For example, if A = {{0, 1, 5}, {1, 6}, {1, 5}}, then

A = {0, 1, 5, 6}, A = {1}.

Two other examples are A∪B =

{A, B},

P A = A.

In cases where we n, the union have a set An for each natural number ofall these sets, {An | n ∈ N}, is usually denoted “ n∈N An ” or just “ n An .” The ordered pair x, y of objects x and y must be deﬁned in such a way that x, y = u, v

iff

x =u

and

y = v.

Any deﬁnition that has this property will do; the standard one is x, y = {{x}, {x, y}}. For ordered triples we deﬁne x, y, z = x, y , z .

4

A Mathematical Introduction to Logic More generally we deﬁne n-tuples recursively by x1 , . . . , xn+1 = x1 , . . . , xn , xn+1 for n > 1. It is convenient to deﬁne also x = x; the preceding equation then holds also for n = 1. S is a ﬁnite sequence (or string) of members of A iff for some positive integer n, we have S = x1 , . . . , xn , where each xi ∈ A. (Finite sequences are often deﬁned to be certain ﬁnite functions, but the above deﬁnition is slightly more convenient for us.) A segment of the ﬁnite sequence S = x1 , . . . , xn is a ﬁnite sequence xk , xk+1 , . . . , xm−1 , xm ,

where

1 ≤ k ≤ m ≤ n.

This segment is an initial segment iff k = 1 and it is proper iff it is different from S. If x1 , . . . , xn = y1 , . . . , yn , then it is easy to see that xi = yi for 1 ≤ i ≤ n. (The proof uses induction on n and the basic property of ordered pairs.) But if x1 , . . . , xm = y1 , . . . , yn , then it does not in general follow that m = n. After all, every ordered triple is also an ordered pair. But we claim that m and n can be unequal only if some xi is itself a ﬁnite sequence of y j ’s, or the other way around: LEMMA 0A Assume that x1 , . . . , xm = y1 , . . . , ym , . . . , ym+k . Then x1 = y1 , . . . , yk+1 . PROOF. We use induction on m. If m = 1, the conclusion is immediate. For the inductive step, assume that x1 , . . . , xm , xm+1 = y1 , . . . , ym+k , ym+1+k . Then the ﬁrst components of this ordered pair must be equal: x1 , . . . , xm = y1 , . . . , ym+k . Now apply the inductive hypothesis. For example, suppose that A is a set such that no member of A is a ﬁnite sequence of other members. Then if x1 , . . . , xm = y1 , . . . , yn and each xi and y j is in A, then by the above lemma m = n. Whereupon we have xi = yi as well. From sets A and B we can form their Cartesian product, the set A × B of all pairs x, y for which x ∈ A and y ∈ B. An is the set of all n-tuples of members of A. For example, A3 = (A × A) × A. A relation R is a set of ordered pairs. For example, the ordering relation on the numbers 0–3 is captured by — and in fact is — the set of ordered pairs { 0, 1 , 0, 2 , 0, 3 , 1, 2 , 1, 3 , 2, 3 }. The domain of R (written dom R) is the set of all objects x such that x, y ∈ R for some y. The range of R (written ran R) is the set of all objects y such that x, y ∈ R for some x. The union of dom R and ran R is the ﬁeld of R, ﬂd R.

Chapter 0:

Useful Facts about Sets

5

An n-ary relation on A is a subset of An . If n > 1, it is a relation. But a 1-ary (unary) relation on A is simply a subset of A. A particularly simple binary relation on A is the equality relation { x, x | x ∈ A} on A. For an n-ary relation R on A and subset B of A, the restriction of R to B is the intersection R ∩ B n . For example, the relation displayed above is the restriction to the set B = {0, 1, 2, 3} of the ordering relation on N. A function is a relation F with the property of being single-valued: For each x in dom F there is only one y such that x, y ∈ F. As usual, this unique y is said to be the value F(x) that F assumes at x. (This notation goes back to Euler. It is a pity he did not choose (x)F instead; that would have been helpful for the composition of functions: f ◦ g is the function whose value at x is f (g(x)), obtained by applying ﬁrst g and then f .) We say that F maps A into B and write F:A→B to mean that F is a function, dom F = A, and ran F ⊆ B. If in addition ran F = B, then F maps A onto B. F is one-to-one iff for each y in ran F there is only one x such that x, y ∈ F. If the pair x, y is in dom F, then we let F(x, y) = F( x, y ). This notation is extended to n-tuples; F(x1 , . . . , xn ) = F( x1 , . . . , xn ). An n-ary operation on A is a function mapping An into A. For example, addition is a binary operation on N, whereas the successor operation S (where S(n) = n + 1) is a unary operation on N. If f is an n-ary operation on A, then the restriction of f to a subset B of A is the function g with domain B n which agrees with f at each point of B n . Thus, g = f ∩ (B n × A). This g will be an n-ary operation on B iff B is closed under f , in the sense that f (b1 , . . . , bn ) ∈ B whenever each bi is in B. In this case, g = f ∩ B n+1 , in agreement with our deﬁnition of the restriction of a relation. For example, the addition operation on N, which contains such triples as 3, 2 , 5 , is the restriction to N of the addition operation on R, which contains many more triples. A particularly simple unary operation on A is the identity function I d on A, given by the equation I d(x) = x

for

x ∈ A.

Thus I d = { x, x | x ∈ A}. For a relation R, we deﬁne the following: R is reﬂexive on A iff x, x ∈ R for every x in A. R is symmetric iff whenever x, y ∈ R, then also y, x ∈ R. R is transitive iff whenever both x, y ∈ R and y, z ∈ R (if this ever happens), then also x, z ∈ R.

6

A Mathematical Introduction to Logic R satisﬁes trichotomy on A iff for every x and y in A, exactly one of the three possibilities, x, y ∈ R, x = y, or y, x ∈ R, holds. R is an equivalence relation on A iff R is a binary relation on A that is reﬂexive on A, symmetric, and transitive. R is an ordering relation on A iff R is transitive and satisﬁes trichotomy on A. For an equivalence relation R on A we deﬁne, for x ∈ A, the equivalence class [x] of x to be {y | x, y ∈ R}. The equivalence classes then partition A. That is, the equivalence classes are subsets of A such that each member of A belongs to exactly one equivalence class. For x and y in A, [x] = [y]

iff

x, y ∈ R.

The set N of natural numbers is the set {0, 1, 2, . . .}. (Natural numbers can also be deﬁned set-theoretically, a point that arises brieﬂy in Section 3.7.) A set A is ﬁnite iff there is some one-to-one function f mapping (for some natural number n) the set A onto {0, 1, . . . , n − 1}. (We can think of f as “counting” the members of A.) A set A is countable iff there is some function mapping A one-to-one into N. For example, any ﬁnite set is obviously countable. Now consider an inﬁnite countable set A. Then from the given function f mapping A one-to-one into N, we can extract a function f mapping A one-toone onto N. For some a0 ∈ A, f (a0 ) is the least member of ran f , let f (a0 ) = 0. In general there is a unique an ∈ A such that f (an ) is the (n + 1)st member of ran f ; let f (an ) = n. Note that A = {a0 , a1 , . . .}. (We can also think of f as “counting” the members of A, only now the counting process is inﬁnite.) THEOREM 0B Let A be a countable set. Then the set of all ﬁnite sequences of members of A is also countable. PROOF. The set S of all such ﬁnite sequences can be characterized by the equation An+1 . S= n∈N

Since A is countable, we have a function f mapping A oneto-one into N. The basic idea is to map S one-to-one into N by assigning to a0 , a1 , . . . , am the number 2 f (a0 )+1 3 f (a1 )+1 · . . . · pmf (am )+1 , where pm is the (m + 1)st prime. This suffers from the defect that this assignment might not be well-deﬁned. For conceivably there could be a0 , a1 , . . . , am = b0 , b1 , . . . , bn , with ai and

n. But this is not serious; just assign to b j in A but with m = each member of S the smallest number obtainable in the above

Chapter 0:

Useful Facts about Sets

7

fashion. This gives us a well-deﬁned map; it is easy to see that it is one-to-one. At times we will speak of trees, which can be useful in providing intuitive pictures of some situations. But our comments on trees will always be informal; the theorems and proofs will not rely on trees. Accordingly, our discussion here of trees will be informal. For each tree there is an underlying ﬁnite partial ordering. We can draw a picture of this partial ordering R; if a, b ∈ R, then we put a lower than b and connect the points by a line. Pictures of two typical tree orderings are shown.

(In mathematics, trees grow downward, not upward.) There is always a highest point in the picture (the root). Furthermore, while branching is permitted below some vertex, the points above any given vertex must lie along a line. In addition to this underlying ﬁnite partial ordering, a tree also has a labeling function whose domain is the set of vertices. For example, one tree, in which the labels are natural numbers, is shown.

At a few points in the book we will use the axiom of choice. But usually these uses can be eliminated if the theorems in question are restricted to countable languages. Of the many equivalent statements of the axiom of choice, Zorn’s lemma is especially useful. Say that a collection C of sets is a chain iff for any elements x and y of C, either x ⊆ y or y ⊆ x. ZORN ’S LEMMA Let A be a set such that for any chain C ⊆ A, the set C is in A. Then there is some element m ∈ A which is maximal in the sense that it is not a subset of any other element of A.

8

A Mathematical Introduction to Logic

Cardinal Numbers All inﬁnite sets are big, but some are bigger than others. (For example, the set of real numbers is bigger than the set of integers.) Cardinal numbers provide a convenient, although not indispensable, way of talking about the size of sets. It is natural to say that two sets A and B have the same size iff there is a function that maps A one-to-one onto B. If A and B are ﬁnite, then this concept is equivalent to the usual one: If you count the members of A and the members of B, then you get the same number both times. But it is applicable even to inﬁnite sets A and B, where counting is difﬁcult. Formally, then, say that A and B are equinumerous (written A ∼ B) iff there is a one-to-one function mapping A onto B. For example, the set N of natural numbers and the set Z of integers are equinumerous. It is easy to see that equinumerosity is reﬂexive, symmetric, and transitive. For ﬁnite sets we can use natural numbers as measures of size. The same natural number would be assigned to two ﬁnite sets (as measures of their size) iff the sets were equinumerous. Cardinal numbers are introduced to enable us to generalize this situation to inﬁnite sets. To each set A we can assign a certain object, the cardinal number (or cardinality) of A (written card A), in such a way that two sets are assigned the same cardinality iff they are equinumerous: card A = card B

iff

A ∼ B.

(K)

There are several ways of accomplishing this; the standard one these days takes card A to be the least ordinal equinumerous with A. (The success of this deﬁnition relies on the axiom of choice.) We will not discuss ordinals here, since for our purposes it matters very little what card A actually is, any more than it matters what the number 2 actually is. What matters most is that (K) holds. It is helpful, however, if for a ﬁnite set A, card A is the natural number telling how many elements A has. Something is a cardinal number, or simply a cardinal, iff it is card A for some set A. (Georg Cantor, who ﬁrst introduced the concept of cardinal number, characterized in 1895 the cardinal number of a set M as “the general concept which, with the help of our active intelligence, comes from the set M upon abstraction from the nature of its various elements and from the order of their being given.”) Say that A is dominated by B (written A B) iff A is equinumerous with a subset of B. In other words, A B iff there is a one-to-one function mapping A into B. The companion concept for cardinals is card A ≤ card B

iff

A B.

(It is easy to see that ≤ is well deﬁned; that is, whether or not κ ≤ λ depends only on the cardinals κ and λ themselves, and not the choice of

Chapter 0:

Useful Facts about Sets

9

sets having these cardinalities.) Dominance is reﬂexive and transitive. A set A is dominated by N iff A is countable. The following is a standard result in this subject. SCHRODER ¨ –BERNSTEIN THEOREM (a) For any sets A and B, if A B and B A, then A ∼ B. (b) For any cardinal numbers κ and λ, if κ ≤ λ and λ ≤ κ, then κ = λ. Part (b) is a simple restatement of part (a) in terms of cardinal numbers. The following theorem, which happens to be equivalent to the axiom of choice, is stated in the same dual manner. THEOREM 0C (a) For any sets A and B, either A B or B A. (b) For any cardinal numbers κ and λ, either κ ≤ λ or λ ≤ κ. Thus of any two cardinals, one is smaller than the other. (In fact, any nonempty set of cardinal numbers contains a smallest member.) The smallest cardinals are those of ﬁnite sets: 0, 1, 2, . . . . There is next the smallest inﬁnite cardinal, card N, which is given the name ℵ0 . Thus we have 0, 1, 2, . . . , ℵ0 , ℵ1 , . . . , where ℵ1 is the smallest cardinal larger than ℵ0 . The cardinality of the real numbers, card R, is called “2ℵ0 .” Since R is uncountable, we have ℵ0 < 2ℵ0 . The operations of addition and multiplication, long familiar for ﬁnite cardinals, can be extended to all cardinals. To compute κ + λ we choose disjoint sets A and B of cardinality κ and λ, respectively. Then κ + λ = card(A ∪ B). This is well deﬁned; i.e., κ + λ depends only on κ and λ, and not on the choice of the disjoint sets A and B. For multiplication we use κ · λ = card(A × B). Clearly these deﬁnitions are correct for ﬁnite cardinals. The arithmetic of inﬁnite cardinals is surprisingly simple (with the axiom of choice). The sum or product of two inﬁnite cardinals is simply the larger of them: CARDINAL ARITHMETIC THEOREM For cardinal numbers κ and λ, if κ ≤ λ and λ is inﬁnite, then κ + λ = λ. Furthermore, if κ =

0, then κ · λ = λ. In particular, for inﬁnite cardinals κ, ℵ0 · κ = κ.

10

A Mathematical Introduction to Logic THEOREM 0D For an inﬁnite set A, the set n An+1 of all ﬁnite sequences of elements of A has cardinality equal to card A. We already proved this for the case of a countable A (see Theorem 0B). PROOF. Each An+1 has cardinality equal to card A, by the cardinal arithmetic theorem (applied n times). So we have the union of ℵ0 sets of this size, yielding ℵ0 · card A = card A points altogether. EXAMPLE. It follows that the set of algebraic numbers has cardinality ℵ0 . First, we can identify each polynomial (in one variable) over the integers with the sequence of its coefﬁcients. Then by the theorem there are ℵ0 polynomials. Each polynomial has a ﬁnite number of roots. To give an extravagant upper bound, note that even if each polynomial had ℵ0 roots, we would then have ℵ0 · ℵ0 = ℵ0 algebraic numbers altogether. Since there are at least this many, we are done. Since there are uncountably many (in fact, 2ℵ0 ) real numbers, it follows that there are uncountably many (in fact, 2ℵ0 ) transcendental numbers.

Chapter

O N E

Sentential Logic SECTION 1.0 Informal Remarks on Formal Languages We are about to construct (in the next section) a language into which we can translate English sentences. Unlike natural languages (such as English or Chinese), it will be a formal language, with precise formation rules. But before the precision begins, we will consider here some of the features to be incorporated into the language. As a ﬁrst example, the English sentence “Traces of potassium were observed” can be translated into the formal language as, say, the symbol K. Then for the closely related sentence “Traces of potassium were not observed,” we can use (¬ K). Here ¬ is our negation symbol, read as “not.” One might also think of translating “Traces of potassium were not observed” by some new symbol, for example, J, but we will prefer instead to break such a sentence down into atomic parts as much as possible. For an unrelated sentence, “The sample contained chlorine,” we choose, say, the symbol C. Then the following compound English sentences can be translated as the formulas shown at the right: If traces of potassium were observed, then the sample did not contain chlorine. The sample contained chlorine, and traces of potassium were observed.

(K → (¬ C))

(C ∧ K)

11

12

A Mathematical Introduction to Logic The second case uses our conjunction symbol ∧ to translate “and.” The ﬁrst one uses the more familiar arrow to translate “if . . . , then . . . .” In the following example the disjunction symbol ∨ is used to translate “or”: Either no traces of potassium were observed, or the sample did not contain chlorine. Neither did the sample contain chlorine, nor were traces of potassium observed.

((¬ K) ∨ (¬ C)) (¬ (C ∨ K)) or ((¬ C) ∧ (¬ K))

For this last sentence, we have two alternative translations. The relationship between them will be discussed later. One important aspect of the decompositions we will make of compound sentences is that whenever we are given the truth or falsity of the atomic parts, we can then immediately calculate the truth or falsity of the compound. Suppose, for example, that the chemist emerges from her laboratory and announces that she observed traces of potassium but that the sample contained no chlorine. We then know that the four above sentences are true, false, true, and false, respectively. In fact, we can construct in advance a table giving the four possible experimental results (Table I). We will return to the discussion of such tables in Section 1.2. TABLE I K

C

(¬(C ∨ K))

((¬ C) ∧ (¬ K))

F F T T

F T F T

T F F F

T F F F

Use of formal languages will allow us to escape from the imprecision and ambiguities of natural languages. But this is not done without cost; our formal languages will have a sharply limited degree of expressiveness. In order to describe a formal language we will generally give three pieces of information: 1. We will specify the set of symbols (the alphabet). In the present case of sentential logic some of the symbols are (, ), →, ¬, A1 , A2 , . . . . 2. We will specify the rules for forming the “grammatically correct” ﬁnite sequences of symbols. (Such sequences will be called well-formed formulas or wffs.) For example, in the present case (A1 → (¬ A2 ))

Chapter 1:

Sentential Logic

13

will be a wff, whereas )) → A3 will not. 3. We will also indicate the allowable translations between English and the formal language. The symbols A1 , A2 , . . . can be translations of declarative English sentences. Only in this third part is any meaning assigned to the wffs. This process of assigning meaning guides and motivates all we do. But it will also be observed that it would theoretically be possible to carry out various manipulations with wffs in complete ignorance of any possible meaning. A person aware of only the ﬁrst two pieces of information listed above could perform some of the things we will do, but it would make no sense to him. Before proceeding, let us look brieﬂy at another class of formal languages of widespread interest today. These are the languages used by (or at least in connection with) digital computers. There are many of these languages. In one of them a typical wff is 011010110101000111110001000001111010. In another a typical wff is STEP#ADDIMAX, A. (Here # is a symbol called a blank; it is brought into the alphabet so that a wff will be a string of symbols.) A well-known language called C++ has wffs such as while(∗s++); In all cases there is a given way of translating the wffs into English, and (for a restricted class of English sentences) a way of translating from English into the formal language. But the computer is unaware of the English language. An unthinking automaton, the computer juggles symbols and follows its program slavishly. We could approach formal languages that way, too, but it would not be much fun.

SECTION 1.1 The Language of Sentential Logic We assume we are given an inﬁnite sequence of distinct objects which we will call symbols, and to which we now give names (Table II). We further assume that no one of these symbols is a ﬁnite sequence of other symbols.

14

A Mathematical Introduction to Logic TABLE II Symbol

Verbose name

Remarks

( ) ¬ ∧ ∨ → ↔ A1 A2 ··· An ···

left parenthesis right parenthesis negation symbol conjunction symbol disjunction symbol conditional symbol biconditional symbol ﬁrst sentence symbol second sentence symbol

punctuation punctuation English: not English: and English: or (inclusive) English: if , then English: if and only if

nth sentence symbol

Several remarks are now in order: 1. The ﬁve symbols ¬, ∧, ∨, →, ↔ are called sentential connective symbols. Their use is suggested by the English translation given above. The sentential connective symbols, together with the parentheses, are the logical symbols. In translating to and from English, their role never changes. The sentence symbols are the parameters (or nonlogical symbols). Their translation is not ﬁxed; instead they will be open to a variety of interpretations, as we shall illustrate shortly. 2. We have included inﬁnitely many sentence symbols. On the one hand, a more modest alternative would be to have one sentence symbol A, and a prime . Then we could use the potentially inﬁnite sequence A, A , A , . . . in place of A1 , A2 , A3 . . . . This alternative has the advantage that it brings the total number of different symbols down to nine. On the other hand, a less modest alternative would be to allow an arbitrary set of set sentence symbols, countable or not. Much of what is said in this chapter would continue to be applicable in this case; the exceptions are primarily in Section 1.7. 3. Some logicians prefer to call An the nth proposition symbol (and to speak of propositional logic instead of sentential logic). This stems

Chapter 1:

Sentential Logic

15

from wanting the word “sentence” to refer to a particular utterance, and wanting a proposition to be that which a sentence asserts. 4. We call these objects “symbols,” but we remain neutral as to what the exact ontological status of symbols might be. In the leftmost column of our list of symbols, names of the symbols are printed. For example, A243 is one symbol, namely the 243rd sentence symbol. (On the other hand, “A243 ” is a name of that symbol. The conditional symbol may or may not have the geometric property of being shaped like an arrow, although its name “→” does.) The symbols themselves can be sets, numbers, marbles, or objects from a universe of linguistic objects. In the last case, it is conceivable that they are actually the same things as the names we use for them. Another possibility, which will be explained in the next chapter, is that the sentence symbols are themselves formulas in another language. 5. We have assumed that no symbol is a ﬁnite sequence of other symbols. We mean by this that not only are the symbols listed distinct (e.g., A3 =

↔), but no one of them is a ﬁnite sequence of two or more

¬, A4 , ( . The purpose of symbols. For example, we demand that A3 = this assumption is to assure that ﬁnite sequences of symbols are uniquely decomposable. If a1 , . . . , am = b1 , . . . , bn and each ai and each b j is a symbol, then m = n and ai = bi . (See Chapter 0, Lemma 0A, and subsequent remarks.) An expression is a ﬁnite sequence of symbols. We can specify an expression by concatenating the names of the symbols; thus (¬ A1 ) is the sequence (, ¬, A1 , ) . This notation is extended: If α and β are sequences of symbols, then αβ is the sequence consisting ﬁrst of the symbols in the sequence α followed by the symbols in the sequence β. For example, if α and β are the expressions given by the equations α = (¬ A1 ), β = A2 , then (α → β) is the expression ((¬ A1 ) → A2 ). We should now look at a few examples of possible translations of English sentences into expressions of the formal language. Let A, B, . . . , Z be the ﬁrst twenty-six sentence symbols. (For example, E = A5 .) 1. English: The suspect must be released from custody. Translation: R. English: The evidence obtained is admissible. Translation: E. English: The evidence obtained is inadmissible. Translation: (¬ E).

16

A Mathematical Introduction to Logic English: The evidence obtained is admissible, and the suspect need not be released from custody. Translation: (E ∧ (¬ R)). English: Either the evidence obtained is admissible, or the suspect must be released from custody (or possibly both). Translation: (E ∨ R). English: Either the evidence obtained is admissible, or the suspect must be released from custody, but not both. Translation: ((E ∨ R) ∧ (¬ (E ∧ R))). We intend always to use the symbol ∨ to translate the word “or” in its inclusive meaning of “and/or.” English: The evidence obtained is inadmissible, but the suspect need not be released from custody. Translation: ((¬ E) ∧ (¬ R)). On the other hand, the expression ((¬ E) ∨ (¬ R)) translates the English: Either the evidence obtained is inadmissible or the suspect need not be released from custody. 2. English: If wishes are horses, then beggars will ride. Translation: (W → B). English: Beggars will ride if and only if wishes are horses. Translation: (B ↔ W). 3. English: This commodity constitutes wealth if and only if it is transferable, limited in supply, and either productive of pleasure or preventive of pain. Translation: (W ↔ (T ∧ (L ∧ (P ∨ Q)))). Here W is the translation of “This commodity constitutes wealth.” Of course in the preceding example we used W to translate a different sentence. We are not tied to any one translation. One note of caution: Do not confuse a sentence in the English language (Roses are red) with a translation of that sentence in the formal language (such as R). These are different. The English sentence is presumably either true or false. But the formal expression is simply a sequence of symbols. It may indeed be interpreted in one context as a true (or false) English sentence, but it can have other interpretations in other contexts. Now some expressions cannot be obtained as translations of any English sentences but are mere nonsense, such as ((→ A3 . We want to deﬁne the well-formed formulas (wffs) to be the “grammatically correct” expressions; the nonsensical expressions must be excluded. The deﬁnition will have the following consequences: (a) Every sentence symbol is a wff. (b) If α and β are wffs, then so are (¬ α), (α ∧ β), (α ∨ β), (α → β), and (α ↔ β). (c) No expression is a wff unless it is compelled to be one by (a) and (b).

Chapter 1:

Sentential Logic

17

We want to make this third property (about compulsion) more precise. A well-formed formula (or simply formula or wff ) is an expression that can be built up from the sentence symbols by applying some ﬁnite number of times the formula-building operations (on expressions) deﬁned by the equations E¬ (α) = (¬ α) E∧ (α, β) = (α ∧ β), E∨ (α, β) = (α ∨ β), E→ (α, β) = (α → β), E↔ (α, β) = (α ↔ β). For example, ((A1 ∧ A10 ) → ((¬ A3 ) ∨ (A8 ↔ A3 ))) is a wff, as can be seen by contemplating its ancestral tree:

The tree illustrates the construction of the expression from four sentence symbols by ﬁve applications of formula-building operations. This example is atypical in that it uses all ﬁve formula-building operations. For a smaller example, A3 is a wff; its ancestral tree has only a single vertex; each formula-building operation is used zero times. But this is as small as examples get; we do not count the empty sequence as being “built up from the sentence symbols.” This sort of construction, of taking some basic building blocks (here the sentence symbols) and “closing” under some operations (here ﬁve operations), occurs frequently both in logic and in other branches of mathematics. In Section 1.4, we will examine this sort of construction in a more general setting. We can elaborate on the “building” idea as follows. Deﬁne a construction sequence to be a ﬁnite sequence ε1 , . . . , εn of expressions

18

A Mathematical Introduction to Logic such that for each i ≤ n we have at least one of εi is a sentence symbol εi = E¬ (ε j ) for some j < i εi = E (ε j , εk ) for some j < i, k < i where is one of the binary connectives, ∧, ∨, →, ↔. Then the wellformed formulas can be characterized as the expressions α such that some construction sequence ends with α. We can think of εi as the expression at stage i in the building process. For our earlier example, ((A1 ∧ A10 ) → ((¬ A3 ) ∨ (A8 ↔ A3 ))) we obtain a construction sequence by squashing its ancestral tree into a linear ordering. One feature of this sort of construction is that it yields an induction principle. Say that a set S is closed under a two-place function f iff whenever x ∈ S and y ∈ S then f (x, y) ∈ S, and similarly for one-place functions and so forth. INDUCTION PRINCIPLE If S is a set of wffs containing all the sentence symbols and closed under all ﬁve formula-building operations, then S is the set of all wffs. FIRST PROOF. Consider an arbitrary wff α. It is built up from sentence symbols by applying some ﬁnite number of times the formulabuilding operations. Working our way up the corresponding ancestral tree, we ﬁnd that each expression in the tree belongs to S. Eventually (that is, after a ﬁnite number of steps) at the top of the tree we ﬁnd that α ∈ S. SECOND PROOF. We will repeat the argument, but without the trees. Consider an arbitrary wff α. It is the last member of some construction sequence ε1 , . . . , εn . By ordinary strong numerical induction on the number i, we see that each εi ∈ S, i ≤ n. That is, we suppose, as our inductive hypothesis, that ε j ∈ S for all j < i. We then verify that εi ∈ S, by considering the various cases. So by strong induction on i, it follows that εi ∈ S for each i ≤ n. In particular, the last member α belongs to S. This principle will receive much use in the coming pages. In the following example we use it to show that certain expressions are not wffs. EXAMPLE. Any expression with more left parentheses than right parentheses is not a wff.

Chapter 1:

Sentential Logic

19

PROOF. The idea is that we start with sentence symbols (having zero left parentheses and zero right parentheses), and then apply formula-building operations which add parentheses only in matched pairs. We can rephrase this argument as follows: The set of “balanced” wffs (having equal numbers of left and right parentheses) contains all sentence symbols and is closed under the formula-building operations. The induction principle then assures us that all wffs are balanced. A special feature of our particular formula-building operations is that they build up and never down. That is, the expression E (α, β) always includes as a segment the entire sequence α (and the entire sequence β) plus other symbols. In particular, it is longer than either α or β. This special feature will simplify the problem of determining, given a wff ϕ, exactly how it was built up. All the building blocks, so to speak, are included as segments in the sequence ϕ. For example, if ϕ does not contain the symbol A4 , then ϕ can be built up without ever using A4 . (See Exercise 4.)

Exercises 1. Give three sentences in English together with translations into our formal language. The sentences should be chosen so as to have an interesting structure, and the translations should each contain 15 or more symbols. 2. Show that there are no wffs of length 2, 3, or 6, but that any other positive length is possible. 3. Let α be a wff; let c be the number of places at which binary connective symbols (∧, ∨, →, ↔) occur in α; let s be the number of places at which sentence symbols occur in α. (For example, if α is (A → (¬ A)) then c = 1 and s = 2.) Show by using the induction principle that s = c + 1. 4. Assume we have a construction sequence ending in ϕ, where ϕ does not contain the symbol A4 . Suppose we delete all the expressions in the construction sequence that contain A4 . Show that the result is still a legal construction sequence. 5. Suppose that α is a wff not containing the negation symbol ¬. (a) Show that the length of α (i.e., the number of symbols in the string) is odd. (b) Show that more than a quarter of the symbols are sentence symbols. Suggestion: Apply induction to show that the length is of the form 4k + 1 and the number of sentence symbols is k + 1.

20

A Mathematical Introduction to Logic

SECTION 1.2 Truth Assignments We want to deﬁne what it means for one wff of our language to follow logically from other wffs. For example, A1 should follow from (A1 ∧ A2 ). For no matter how the parameters A1 and A2 are translated back into English, if the translation of (A1 ∧ A2 ) is true, then the translation of A1 must be true. But the notion of all possible translations back into English is unworkably vague. Luckily the spirit of this notion can be expressed in a simple and precise way. Fix once and for all a set {F, T } of truth values consisting of two distinct points: F, T,

called falsity, called truth.

(It makes no difference what these points themselves are; they might as well be the numbers 0 and 1.) Then a truth assignment v for a set S of sentence symbols is a function v : S → {F, T } assigning either T or F to each symbol in S. These truth assignments will be used in place of the translations into English mentioned in the preceding paragraph. (At this point we have committed ourselves to two-valued logic. It is also possible to study three-valued logic, in which case one has a set of three possible truth values. And then, of course, it is a small additional step to allow 512 or ℵ0 truth values; or to take as the set of truth values the unit interval [0, 1] or some other convenient space. A particularly interesting case is that for which the truth values form something called a complete Boolean algebra. But it is two-valued logic that has always had the greatest signiﬁcance, and we will conﬁne ourselves to this case.) Let S be the set of wffs that can be built up from S by the ﬁve formula-building operations. (S can also be characterized as the set of wffs whose sentence symbols are all in S; see the remarks at the end of the preceding section.) We want an extension v of v, v : S → {F, T }, that assigns the correct truth value to each wff in S. It should meet the following conditions: 0. For any A ∈ S, v(A) = v(A). (Thus v is an extension of v.) For any α, β in S: T if v(α) = F, 1. v((¬ α)) = F otherwise.

Chapter 1:

Sentential Logic 2. v((α ∧ β)) =

21 T F

if v(α) = T otherwise.

and v(β) = T,

T if v(α) = T or v(β) = T (or both), F otherwise. F if v(α) = T and v(β) = F, 4. v((α → β)) = T otherwise. T if v(α) = v(β), 5. v((α ↔ β)) = F otherwise. 3. v((α ∨ β)) =

Conditions 1–5 are given in tabular form in Table III. It is at this point that the intended meaning of, for example, the conjunction symbol enters into our formal proceedings. Note especially the intended meaning of →. Whenever α is assigned F, then (α → β) is considered “vacuously true” and is assigned the value T . For this and the other connectives, it is certainly possible to question how accurately we have reﬂected the common meaning in everyday speech of “if . . . , then,” “or,” etc. But our ultimate concern lies more with mathematical statements than with the subtle nuances of everyday speech. TABLE III α

β

(¬ α)

(α ∧ β)

(α ∨ β)

(α → β)

(α ↔ β)

T T F F

T F T F

F F T T

T F F F

T T T F

T F T T

T F F T

For example, we might translate the English sentence, “If you’re telling the truth then I’m a monkey’s uncle,” by the formula (V → M). We assign this formula the value T whenever you are ﬁbbing. In assigning the value T , we are certainly not asserting any causal connection between your veracity and any simian features of my nephews or nieces. The sentence in question is a conditional statement. It makes an assertion about my relatives provided a certain condition — that you are telling the truth — is met. Whenever that condition fails, the statement is vacuously true. Very roughly, we can think of a conditional formula (α → β) as expressing a promise that if a certain condition is met (viz., that α is true), then β is true. If the condition α turns out not to be met, then the promise stands unbroken, regardless of β. As an example of the calculation of v, let α be the wff ((A2 → (A1 → A6 )) ↔ ((A2 ∧ A1 ) → A6 ))

22

A Mathematical Introduction to Logic and let v be the truth assignment for {A1 , A2 , A6 } such that v(A1 ) = T, v(A2 ) = T, v(A6 ) = F. We want to compute v(α). We can look at the tree which displays the construction of α:

Working from the bottom up, we can assign to each vertex β of the tree the value v(β). So as a ﬁrst step we compute v((A1 → A6 )) = F

and v((A2 ∧ A1 )) = T.

Next we compute v((A2 → (A1 → A6 ))) = F, and so forth. Finally, at the top of the tree we arrive at v(α) = T . Actually this computation can be carried out with much less writing. First, the tree can be given in starker form:

And even this can be compressed into a single line (with the parentheses restored):

Chapter 1:

Sentential Logic

23

THEOREM 12A For any truth assignment v for a set S there is a unique function v : S → {F, T } meeting the aforementioned conditions 0–5. The full proof of this theorem will emerge in the next two sections (1.3 and 1.4). But it should already seem extremely plausible, especially in light of the preceding example. In proving the existence of v, the crucial issue will, in effect, be the uniqueness of the trees mentioned in the example. We say that a truth assignment v satisﬁes ϕ iff v(ϕ) = T . (Of course for this to happen, the domain of v must contain all sentence symbols in ϕ.) Now consider a set of wffs (thought of as hypotheses) and another wff τ (thought of as a possible conclusion). DEFINITION. tautologically implies τ (written |= τ ) iff every truth assignment for the sentence symbols in and τ that satisﬁes every member of also satisﬁes τ . This deﬁnition reﬂects our intuitive feeling that a conclusion follows from a set of hypotheses if the assumption that the hypotheses are true guarantees that the conclusion is true. Several special cases of the concept of tautological implication deserve mention. First, take the special case in which is the empty set ∅. Observe that it is vacuously true that any truth assignment satisﬁes every member of ∅. (How could this fail? Only if there was some unsatisﬁed member of ∅, which is absurd.) Hence we are left with: ∅ |= τ iff every truth assignment (for the sentence symbols in τ ) satisﬁes τ . In this case we say that τ is a tautology (written |= τ ). For example, the wff ((A2 → (A1 → A6 )) ↔ ((A2 ∧ A1 ) → A6 )) was in a recent example found to be satisﬁed by one of the eight possible truth assignments for {A1 , A2 , A6 }. In fact, it is satisﬁed by the other seven as well, and hence is a tautology. Another special case is that in which no truth assignment satisﬁes every member of . Then for any τ , it is vacuously true that |= τ . For example, {A, (¬ A)} |= B. There is no deep principle involved here; it is simply a by-product of our deﬁnitions. EXAMPLE. {A, (A → B)} |= B. There are four truth assignments for {A, B}. It is easy to check that only one of these four satisﬁes both A and (A → B), namely the v for which v(A) = v(B) = T . This v also satisﬁes B.

24

A Mathematical Introduction to Logic If is singleton {σ }, then we write “σ |= τ ” in place of “{σ } |= τ .” If both σ |= τ and τ |= σ , then σ and τ are said to be tautologically equivalent (written σ |==| τ ). For example, in Section 1.0 we encountered the wffs (¬ (C∨K)) and ((¬ C)∧(¬ K)) as alternative translations of an English sentence. We can now assert that they are tautologically equivalent. We can already state here a nontrivial fact whose proof will be given later (in Section 1.7). COMPACTNESS THEOREM Let be an inﬁnite set of wffs such that for any ﬁnite subset 0 of , there is a truth assignment that satisﬁes every member of 0 . Then there is a truth assignment that satisﬁes every member of . This theorem can be restated more simply as: If every ﬁnite subset of is satisﬁable, then itself is satisﬁable. (Readers familiar with some general topology might try to discover why this is called “compactness theorem”; it does assert the compactness of a certain topological space. They can then prove the theorem for themselves, by using Tychonoff’s theorem on product spaces.)

Truth Tables There is a systematic procedure, which we will now illustrate, for checking, given wffs σ1 , . . . , σk , and τ , whether or not {σ1 , . . . , σk } |= τ. In particular (when k = 0), the procedure will, given a wff, decide whether or not it is a tautology. As a ﬁrst example, we can show that (¬ (A ∧ B)) |= ((¬ A) ∨ (¬ B)). To do this, we consider all truth assignments for {A, B}. There are four such assignments; in general there are 2n truth assignments for a set of n sentence symbols. The four can be listed in a table: A T T F F

B T F T F

This table can then be expanded to include (¬ (A ∧ B)) and ((¬ A) ∨ (¬ B)). For each formula we compute the T ’s and F’s the way described before, writing the truth value under the correct connective (Table IV). (The two leftmost columns of Table IV are actually unnecessary.) We can now see from this table that all those truth assignments satisfying (¬ (A ∧ B)) — and there are three such — also satisfy ((¬ A) ∨ (¬ B)).

Chapter 1:

Sentential Logic

25 TABLE IV

A

B

T T F F

T F T F

(¬(A ∧ B)) FT TT TF TF

T F F F

T F T F

((¬A) ∨ (¬B)) F F T T

T T F F

FF TT TF TT

T F T F

In fact, the converse holds also, and thus (¬ (A ∧ B)) |==| ((¬ A) ∨ ( ¬ B)). To show that (¬ (A ∧ B)) |= ((¬ A) ∧ (¬ B)) we can construct the table as before. But only one line of the table is needed to establish that there is indeed a truth assignment satisfying (¬ (A ∧ B)) that fails to satisfy ((¬ A) ∧ (¬ B)). The more generally applicable a procedure it is, the less efﬁcient it is likely to be. For example, to show that |= ((A ∨ (B ∧ C)) ↔ ((A ∨ B) ∧ (A ∨ C))), we could apply the truth-table method. But this requires eight lines (for the eight possible truth assignments for {A, B, C}). With a little cleverness, the tedium in this particular case can be reduced:

In the ﬁrst line we assumed only that v(A) = T . Since that is enough information to obtain T for the wff, we assume in all later lines that v(A) = F. In the second line we assume that v(B) = F; this again lets us obtain T for the wff. So we may limit ourselves to the case v(B) = T . Since the expression is symmetric in B and C, we may further suppose v(C) = T . This leaves only the third line, whereupon we are done. As an example of the avoidance of a 16-line table, consider the following tautology:

Here in the ﬁrst line we dispose of the case where v(S) = T . In the second line we dispose of the case where either v(P) = F or v(Q) = F.

26

A Mathematical Introduction to Logic The third line incorporates the two remaining possibilities; here R is the truth value assigned to R and R is the opposite value. For the above example it is possible to see directly why it is a tautology. The stronger the antecedent (the expression on the left side), the weaker the conditional. Thus (P ∧ Q) |= P, (P → R) |= ((P ∧ Q) → R), (((P ∧ Q) → R) → S) |= ((P → R) → S). The problem of developing effective procedures that reduce the tedium is important for theorem proving by machine. Some of the programs here may require testing wffs of sentential logic having thousands of sentence symbols. Truth tables are far too cumbersome for anything this large. The problem of developing highly efﬁcient methods is a current area of research in computer science. Digression. Applying the truth-table method — carried out in full — to a wff with n sentence symbols requires making a table of 2n lines. The trouble is, as n increases, 2n grows “exponentially.” For example, suppose you can generate the table at the rate of a million lines per second. (We are supposing you are aided by a computer, of course.) Then if n = 80, say, you will need “merely” 280 microseconds to form the complete table. How long is that? Converted to years, 280 microseconds is about 38 billion years. For comparison, the age of the universe is around 15 billion years. The conclusion is that 280 microseconds is more time than there has been time! Is there a faster method? Might there be some general method that, given any wff α with n sentence symbols, will determine whether or not α is a tautology in only 10n 5 microseconds (or some other function of n that grows like a polynomial instead of growing exponentially)? (For n = 80, 10n 5 microseconds comes to only 9 hours.) The answer to this question is unknown, but it is widely believed that the answer is negative. The question is called the “P versus NP” problem, the most prominent open problem in theoretical computer science today.

A Selected List of Tautologies 1. Associative and commutative laws for ∧, ∨, ↔. 2. Distributive laws: ((A ∧ (B ∨ C)) ↔ ((A ∧ B) ∨ (A ∧ C))). ((A ∨ (B ∧ C)) ↔ ((A ∨ B) ∧ (A ∨ C))).

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3. Negation: ((¬ (¬ A)) ↔ A). ((¬ (A → B)) ↔ (A ∧ (¬ B))). ((¬ (A ↔ B)) ↔ ((A ∧ (¬ B)) ∨ ((¬ A) ∧ B))). De Morgan’s laws: ((¬ (A ∧ B)) ↔ ((¬ A) ∨ (¬ B))). ((¬ (A ∨ B)) ↔ ((¬ A) ∧ (¬ B))). 4. Other Excluded middle:

(A ∨ (¬ A)).

Contradiction: Contraposition:

(¬ (A ∧ (¬ A))). ((A → B) ↔ ((¬ B) → (¬ A))).

Exportation:

(((A ∧ B) → C) ↔ (A → (B → C))).

Exercises 1. Show that neither of the following two formulas tautologically implies the other: (A ↔ (B ↔ C)), ((A ∧ (B ∧ C)) ∨ ((¬ A) ∧ ((¬ B) ∧ (¬ C)))). 2.

3.

4.

5.

Suggestion: Only two truth assignments are needed, not eight. (a) Is (((P → Q) → P) → P) a tautology? (b) Deﬁne σk recursively as follows: σ0 = (P → Q) and σk+1 = (σk → P). For which values of k is σk a tautology? (Part (a) corresponds to k = 2.) (a) Determine whether or not ((P → Q) ∨ (Q → P)) is a tautology. (b) Determine whether or not ((P∧Q)→R) tautologically implies ((P → R) ∨ (Q → R)). Show that the following hold: (a) ; α |= β iff |= (α → β). (b) α |==| β iff |= (α ↔ β). (Recall that ; α = ∪{α}, the set together with the one possibly new member α.) Prove or refute each of the following assertions: (a) If either |= α or |= β, then |= (α ∨ β). (b) If |= (α ∨ β), then either |= α or |= β.

6. (a) Show that if v1 and v2 are truth assignments which agree on all the sentence symbols in the wff α, then v 1 (α) = v 2 (α). Use the induction principle.

28

A Mathematical Introduction to Logic (b) Let S be a set of sentence symbols that includes those in and τ (and possibly more). Show that |= τ iff every truth assignment for S which satisﬁes every member of also satisﬁes τ . (This is an easy consequence of part (a). The point of part (b) is that we do not need to worry about getting the domain of a truth assignment exactly perfect, as long as it is big enough. For example, one option would be always to use truth assignments on the set of all sentence symbols. The drawback is that these are inﬁnite objects, and there are a great many — uncountably many — of them.) 7. You are in a land inhabited by people who either always tell the truth or always tell falsehoods. You come to a fork in the road and you need to know which fork leads to the capital. There is a local resident there, but he has time only to reply to one yes-or-no question. What one question should you ask so as to learn which fork to take? Suggestion: Make a table. 8. (Substitution) Consider a sequence α1 , α2 , . . . of wffs. For each wff ϕ let ϕ ∗ be the result of replacing the sentence symbol An by αn , for each n. (a) Let v be a truth assignment for the set of all sentence symbols; deﬁne u to be the truth assignment for which u(An ) = v(αn ). Show that u(ϕ) = v(ϕ ∗ ). Use the induction principle. (b) Show that if ϕ is a tautology, then so is ϕ ∗ . (For example, one of our selected tautologies is ((A ∧ B) ↔ (B ∧ A)). From this we can conclude, by substitution, that ((α ∧ β) ↔ (β ∧ α)) is a tautology, for any wffs α and β.) 9. (Duality) Let α be a wff whose only connective symbols are ∧, ∨, and ¬. Let α ∗ be the result of interchanging ∧ and ∨ and replacing each sentence symbol by its negation. Show that α ∗ is tautologically equivalent to (¬ α). Use the induction principle. Remark: It follows that if α |==| β then α ∗ |==| β ∗ . 10. Say that a set 1 of wffs is equivalent to a set 2 of wffs iff for any wff α, we have 1 |= α iff 2 |= α. A set is independent iff no member of is tautologically implied by the remaining members in . Show that the following hold. (a) A ﬁnite set of wffs has an independent equivalent subset. (b) An inﬁnite set need not have an independent equivalent subset. ∗ (c) Let = {σ0 , σ1 , . . .}; show that there is an independent equivalent set . (By part (b), we cannot hope to have ⊆ in general.) 11. Show that a truth assignment v satisﬁes the wff ( · · · (A1 ↔ A2 ) ↔ · · · ↔ An )

Chapter 1:

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iff v(Ai ) = F for an even number of i’s, 1 ≤ i ≤ n. (By the associative law for ↔, the placement of the parentheses is not crucial.) 12. There are three suspects for a murder: Adams, Brown, and Clark. Adams says “I didn’t do it. The victim was an old acquaintance of Brown’s. But Clark hated him.” Brown states “I didn’t do it. I didn’t even know the guy. Besides I was out of town all that week.” Clark says “I didn’t do it. I saw both Adams and Brown downtown with the victim that day; one of them must have done it.” Assume that the two innocent men are telling the truth, but that the guilty man might not be. Who did it? 13. An advertisement for a tennis magazine states, “If I’m not playing tennis, I’m watching tennis. And if I’m not watching tennis, I’m reading about tennis.” We can assume that the speaker cannot do more than one of these activities at a time. What is the speaker doing? (Translate the given sentences into our formal language; consider the possible truth assignments.) 14. Let S be the set of all sentence symbols, and assume that v : S → {F, T } is a truth assignment. Show there is at most one extension v meeting conditions 0–5 listed at the beginning of this section. (Suppose that v 1 and v 2 are both such extensions. Use the induction principle to show that v 1 = v 2 .) 15. Of the following three formulas, which tautologically imply which? (a) (A ↔ B) (b) (¬ ((A → B) → (¬ (B → A)))) (c) (((¬ A) ∨ B) ∧ (A ∨ (¬ B)))

SECTION 1.3 A Parsing Algorithm The purpose of this section is to prove that we have used enough parentheses to eliminate any ambiguity in analyzing wffs. (The existence of the extension v of a truth assignment v will hinge on this lack of ambiguity.1 ) It is instructive to consider the result of not having parentheses at all. The resulting ambiguity is illustrated by the wff A1 ∨ A2 ∧ A3 ,

1

The reader who has already accepted the existence of v may omit almost all of this section. The ﬁnal subsection, on omitting parentheses, will still be needed.

30

A Mathematical Introduction to Logic which can be formed in two ways, corresponding to ((A1 ∨ A2 ) ∧ A3 ) and to (A1 ∨ (A2 ∧ A3 )). If v(A1 ) = T and v(A3 ) = F, then there is an unresolvable conﬂict that arises in trying to compute v(A1 ∨ A2 ∧ A3 ). We must show that with our parentheses this type of ambiguity does not arise but that on the contrary each wff is formed in a unique way. There is one sense in which this fact is unimportant: If it failed, we would simply change notation until it was true. For example, instead of building formulas by means of concatenation, we could have used ordered pairs and triples: ¬, α and α, ∧, β , and so forth. (This is, in fact, a tidy, but untraditional, method.) It would then be immediate that formulas had unique decompositions. But we do not have to resort to this device, and we will now prove that we do not. LEMMA 13A theses. PROOF.

Every wff has the same number of left as right paren-

This was done as an example at the end of Section 1.1.

LEMMA 13B Any proper initial segment of a wff contains an excess of left parentheses. Thus no proper initial segment of a wff can itself be a wff. PROOF. We apply the induction principle to the set S of wffs possessing the desired property (that proper initial segments are leftheavy). A wff consisting of a sentence symbol alone has no proper initial segments and hence is in S vacuously. To verify that S is closed under E∧ , consider α and β in S. The proper initial segments of (α ∧ β) are the following: 1. 2. 3. 4. 5. 6.

(. (α0 , where α0 is a proper initial segment of α. (α. (α∧. (α ∧ β0 , where β0 is a proper initial segment of β. (α ∧ β.

By applying the inductive hypothesis that α and β are in S (in cases 2 and 5), we obtain the desired conclusion. For closure under the other four formula-building operations, the argument is similar.

A Parsing Algorithm We are now going to describe a procedure that, given an expression, both will determine whether or not the expression is a legal wff, and if it is, will construct the tree showing how it was built up from sentence symbols by application of the formula-building operations. Moreover,

Chapter 1:

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31

we will see that this tree is uniquely determined by the wff. It is this last fact that will assure us that we have enough parentheses for an unambiguous notation. Assume then that we are given an expression. We will construct a tree, with the given expression at the top. Initially it is the only vertex in the tree, but as the procedure progresses, the tree will grow downward from the given expression. For an example, imagine the tree in Section 1.1. The algorithm consists of the following four steps: 1. If all minimal vertices (the ones at the bottom) have sentence symbols, then the procedure is completed. (The given expression is indeed a wff, and we have constructed its tree.) Otherwise, select a minimal vertex having an expression that is not a sentence symbol. We examine that expression. 2. The ﬁrst symbol must∗ be (. If the second symbol is the negation symbol, jump to step 4. Otherwise, go on to step 3. 3. Scan the expression from the left until ﬁrst reaching (α, where α is a nonempty expression having a balance between left and right parentheses.∗∗ Then α is the ﬁrst of the two constituents. The next symbol must∗ be ∧, ∨, →, or ↔. This is the principal connective. The remainder of the expression, β), must∗ consist of an expression β and a right parenthesis. We extend the tree by creating two new vertices below the present one, with α as the expression at the “left child” vertex, and β as the expression at the “right child” vertex. Return to step 1. 4. The ﬁrst two symbols are now known to be (¬. The remainder of the expression, β), must* consist of an expression β and a right parenthesis. We extend the tree by creating one new vertex below the present one, with β as the expression at the child vertex. Return to step 1. Now for some comments in support of the correctness of this algorithm. First, given any expression, the procedure halts after a ﬁnite number of steps. This is because any vertex contains a shorter expression than the one above it, so the depth of the tree is bounded by the length of the given expression. Secondly, the choices made by the procedure could not have been made differently. For example, in step 3 we arrive at an expression α. We could not use less than α for a constituent, because it would not have a balance between left and right parentheses (as required by Lemma 13A).

∗ ∗∗

If not, then the expression here is not a wff. We reject the given expression as a non-wff, and halt. If the end of the expression is reached before ﬁnding such an α, then the expression here is not a wff. We reject the given expression as a non-wff, and halt.

32

A Mathematical Introduction to Logic We could not use more than α, because that would have the proper initial segment α that was balanced (violating Lemma 13B). Thus α is forced upon us. And then the choice of the principal connective is inevitable. We conclude that this algorithm constructs the only possible tree for the given expression. Thirdly, if the algorithm uses the footnotes to reject the given expression, then that expression could not have been a wff — the rejection is correct. That is because the only possible attempt to make its tree failed. Finally, if the algorithm does not use the rejection footnotes, then the given expression is indeed a legal wff. That is because we have its tree; by working our way up the tree, we ﬁnd inductively that every vertex has a wff, including the top vertex. The second remark above lets us conclude that our language has enough parentheses; each wff has a unique tree of the sort constructed here. We have “unique readability.” And not only does a unique tree exist for a wff, we know how to construct it; we can carry out this algorithm, using a sufﬁciently large piece of paper. Now to go back to the question of the existence of the extension v of a truth assignment v: It is the uniqueness of the trees that is the crucial fact here. For any wff ϕ there is a unique tree constructing it. By working our way up this tree, assigning a truth value v(α) at each vertex α, we can unambiguously arrive at a value for v(ϕ). And the function described in this way meets conditions 0–5 listed at the start of Section 1.2. And not only that, we know how, given ϕ and the values of v at its sentence symbols, to carry out the calculation of v(ϕ). Thus, using the parsing algorithm, we can make a function v as described in Theorem 12A. And there can be only one such v; cf. Exercise 14 in Section 1.2. The only reason the existence of v is an issue at all is that in Section 1.2 it is described by recursion. That is, v(ϕ) is speciﬁed by making use of the function v itself, applied to smaller formulas. In the next section, we approach the matter of deﬁning a function by recursion more generally. By treating the subject more abstractly, we can better isolate what is at stake.

Polish Notation It is possible to avoid both ambiguity and parentheses. This can be done by a very simple device. Instead of, for example, (α ∧ β) we use ∧αβ. Let the set of P-wffs be the set generated from the sentence symbols by the ﬁve operations D∨ (α, β) = ∨αβ, D¬ (α) = ¬ α, D∧ (α, β) = ∧αβ, D→ (α, β) = →αβ, D↔ (α, β) = ↔αβ.

Chapter 1:

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For example, one P-wff is → ∧ AD ∨ ¬ B ↔ CB. Here the need for an algorithm to analyze the structure is quite apparent. Even for the short example above, it requires some thought to see how it was built up. We will give a unique readability theorem for such expressions in Section 2.3. This way of writing formulas (but with N , K , A, C, and E in place of ¬, ∧, ∨, →, and ↔, respectively) was introduced by the Polish logician Łukasiewicz. The notation is well suited to automatic processing. Computer compiler programs often begin by converting the formulas given them into Polish notation.

Omitting Parentheses Hereafter when naming wffs, we will not feel compelled to mention explicitly every parenthesis. To establish a more compact notation, we now adopt the following conventions: 1. The outermost parentheses need not be explicitly mentioned. For example, when we write “A ∧ B” we are referring to (A ∧ B). 2. The negation symbol applies to as little as possible. For example, ¬ A ∧ B is (¬ A) ∧ B, i.e., ((¬ A) ∧ B). It is not the same as (¬ (A ∧ B)). 3. The conjunction and disjunction symbols apply to as little as possible, given that convention 2 is to be observed. For example, A ∧ B → ¬C ∨ D

((A ∧ B) → ((¬ C) ∨ D)).

is

4. Where one connective symbol is used repeatedly, grouping is to the right: α∧β ∧γ

is

α ∧ (β ∧ γ ),

α→β →γ

is

α → (β → γ ).

It must be admitted that these conventions violate what was said earlier about naming expressions. We can get away with this only because we no longer have any interest in naming expressions that are not wffs.

Exercises 1. Rewrite the tautologies in the “selected list” at the end of Section 1.2, but using the conventions of the present section to minimize the number of parentheses. 2. Give an example of wffs α and β and expressions γ and δ such that (α ∧ β) = (γ ∧ δ) but α =

γ.

34

A Mathematical Introduction to Logic 3. Carry out the argument for Lemma 13B for the case of the operation E¬ . 4. Suppose that we modify our deﬁnition of wff by omitting all right parentheses. Thus instead of ((A ∧ (¬ B)) → (C ∨ D)) we use ((A ∧ (¬ B → (C ∨ D. Show that we still have unique readability (i.e., each wff still has only one possible decomposition). Suggestion: These expressions have the same number of parentheses as connective symbols. 5. The English language has a tendency to use two-part connectives: “both . . . and . . .” “either . . . or . . .” “if . . . , then . . . .” How does this affect unique readability in English? 6. We have given an algorithm for analyzing a wff by constructing its tree from the top down. There are also ways of constructing the tree from the bottom up. This can be done by looking through the formula for innermost pairs of parentheses. Give a complete description of an algorithm of this sort. 7. Suppose that left and right parentheses are indistinguishable. Thus, instead of (α ∨ (β ∧ γ )) we have |α ∨ |β ∧ γ ||. Do formulas still have unique decomposition?

SECTION 1.4 Induction and Recursion1 Induction There is one special type of construction that occurs frequently both in logic and in other branches of mathematics. We may want to construct a certain subset of a set U by starting with some initial elements of U, and applying certain operations to them over and over again. The set we seek will be the smallest set containing the initial elements and closed under the operations. Its members will be those elements of U which can be built up from the initial elements by applying the operations some ﬁnite number of times.

1

On the one hand, the concepts in this section are important, and they arise in many places throughout mathematics. On the other hand, readers may want to postpone — not skip — study of this section.

Chapter 1:

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In the special case of immediate interest to us, U is the set of expressions, the initial elements are the sentence symbols, and the operations are E¬ , E∧ , etc. The set to be constructed is the set of wffs. But we will encounter other special cases later, and it will be helpful to view the situation abstractly here. To simplify our discussion, we will consider an initial set B ⊆ U and a class F of functions containing just two members f and g, where f :U ×U →U

and

g : U → U.

Thus f is a binary operation on U and g is a unary operation. (Actually F need not be ﬁnite; it will be seen that our simpliﬁed discussion here is, in fact, applicable to a more general situation. F can be any set of relations on U , and in Chapter 2 this greater generality will be utilized. But the case discussed here is easier to visualize and is general enough to illustrate the ideas. For a less restricted version, see Exercise 3.) If B contains points a and b, then the set C we wish to construct will contain, for example, b, f (b, b), g(a), f (g(a), f (b, b)), g( f (g(a), f (b, b))). Of course these might not all be distinct. The idea is that we are given certain bricks to work with, and certain types of mortar, and we want C to contain just the things we are able to build. In deﬁning C more formally, we have our choice of two deﬁnitions. We can deﬁne it “from the top down” as follows: Say that a subset S of U is closed under f and g iff whenever elements x and y belong to S, then so also do f (x, y) and g(x). Say that S is inductive iff B ⊆ S and S is closed under f and g. Let C be the intersection of all the inductive subsets of U ; thus x ∈ C iff x belongs to every inductive subset of U. It is not hard to see (and the reader should check) that C is itself inductive. Furthermore, C is the smallest such set, being included in all the other inductive sets. The second (and equivalent) deﬁnition works “from the bottom up.” We want C to contain the things that can be reached from B by applying f and g a ﬁnite number of times. Temporarily deﬁne a construction sequence to be a ﬁnite sequence x1 , . . . , xn of elements of U such that for each i ≤ n we have at least one of xi ∈ B, xi = f (x j , xk ) for some for some xi = g(x j )

j < i, k < i, j < i.

In other words, each member of the sequence either is in B or results from earlier members by applying f or g. Then let C be the set of all points x such that some construction sequence ends with x.

36

A Mathematical Introduction to Logic Let Cn be the set of points x such that some construction sequence of length n ends with x. Then C1 = B,

C1 ⊆ C2 ⊆ C3 ⊆ · · · ,

and C = n Cn . For example, g( f (a, f (b, b))) is in C5 and hence in C , as can be seen by contemplating the tree shown:

We obtain a construction sequence for g( f (a, f (b, b))) by squashing this tree into a linear sequence. EXAMPLES 1. The natural numbers. Let U be the set of all real numbers, and let B = {0}. Take one operation S, where S(x) = x + 1. Then C = {0, 1, 2, . . .}. The set C of natural numbers contains exactly those numbers obtainable from 0 by applying the successor operation repeatedly. 2. The integers. Let U be the set of all real numbers; let B = {0}. This time take two operations, the successor operation S and the predecessor operation P: S(x) = x + 1

and

P(x) = x − 1.

Now C contains all the integers, C = {. . . , −2, −1, 0, 1, 2, . . .}. Notice that there is more than one way of obtaining 2 as a member of C . For 2 is S(S(0)), but it is also S(P(S(S(0)))). 3. The algebraic functions. Let U contain all functions whose domain and range are each sets of real numbers. Let B contain the identity function and all constant functions. Let F contain the operations (on functions) of addition, multiplication, division, and root extraction. Then C is the class of algebraic functions. 4. The well-formed formulas. Let U be the set of all expressions and let B be the set of sentence symbols. Let F contain the ﬁve formula-building operations on expressions: E¬ , E∧ , E∨ , E → , and E↔ . Then C is the set of all wffs.

Chapter 1:

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At this point we should verify that our two deﬁnitions are actually equivalent, i.e., that C = C . To show that C ⊆ C we need only check that C is inductive, i.e., that B ⊆ C and C is closed under the functions. Clearly B = C1 ⊆ C . If x and y are in C , then we can concatenate their construction sequences and append f (x, y) to obtain a construction sequence placing f (x, y) in C . Similarly, C is closed under g. Finally, to show that C ⊆ C we consider a point in C and a construction sequence x0 , . . . , xn for it. By ordinary induction on i, we can see that xi ∈ C , i ≤ n. First x0 ∈ B ⊆ C . For the inductive step we use the fact that C is closed under the functions. Thus we conclude that Cn = C = C = {S | S is inductive}. n

(A parenthetical remark: Suppose our present study is embedded in axiomatic set theory, where the natural numbers are usually deﬁned from the top down. Then our deﬁnition of C (employing ﬁniteness and hence natural numbers) is not really different from our deﬁnition of C . But we are not working within axiomatic set theory; we are working within informal mathematics. And the notion of natural number seems to be a solid, well-understood intuitive concept.) Since C = C , we call the set simply C and refer to it as the set generated from B by the functions in F. We will often want to prove theorems by using the following: INDUCTION PRINCIPLE Assume that C is the set generated from B by the functions in F. If S is a subset of C that includes B and is closed under the functions of F then S = C. PROOF. S is inductive, so C = C ⊆ S. We are given the other inclusion. The special case now of interest to us is, of course, Example 4. Here C is the class of wffs generated from the set of sentence symbols by the formula-building operations. This special case has interesting features of its own. Both α and β are proper segments of E∧ (α, β), i.e., of (α ∧ β). More generally, if we look at the family tree of a wff, we see that each constituent is a proper segment of the end product.

38

A Mathematical Introduction to Logic Suppose, for example, that we temporarily call an expression special if the only sentence symbols in it are among {A2 , A3 , A5 } and the only connective symbols in it are among {¬, →}. Then no special wff requires A9 or E∧ for its construction. In fact, every special wff belongs to the set Cs generated from {A2 , A3 , A5 } by E¬ and E → . (We can use the induction principle to show that every wff either belongs to Cs or is not special.)

Recursion We return now to the more abstract case. There is a set U (such as the set of all expressions), a subset B of U (such as the set of sentence symbols), and two functions f and g, where f :U ×U →U

and

g : U → U.

C is the set generated from B by f and g. The problem we now want to consider is that of deﬁning a function on C recursively. That is, we suppose we are given 1. Rules for computing h(x) for x ∈ B. 2a. Rules for computing h( f (x, y)), making use of h(x) and h(y). 2b. Rules for computing h(g(x)), making use of h(x). (For example, this is the situation discussed in Section 1.2, where h is the extension of a truth assignment for B.) It is not hard to see that there can be at most one function h on C meeting all the given requirements. But it is possible that no such h exists; the rules may be contradictory. For example, let U = the set of real numbers, B = {0}, f (x, y) = x · y, g(x) = x + 1. Then C is the set of natural numbers. Suppose we impose the following requirements on h: 1. h(0) = 0. 2a. h( f (x, y)) = f (h(x), h(y)). 2b. h(g(x)) = h(x) + 2. Then no such function h can exist. (Try computing h(1), noting that we have both 1 = g(0) and 1 = f (g(0), g(0)).) A similar situation is encountered in algebra.2 Suppose that you have a group G that is generated from B by the group multiplication

2

It is hoped that examples such as this will be useful to the reader with some algebraic experience. The other readers will be glad to know that these examples are merely illustrative and not essential to our development of the subject.

Chapter 1:

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39

and inverse operation. Then an arbitrary map of B into a group H is not necessarily extendible to a homomorphism of the entire group G into H . But if G happens to be a free group with set B of independent generators, then any such map is extendible to a homomorphism of the entire group. Say that C is freely generated from B by f and g iff in addition to the requirements for being generated, the restrictions f C and gC of f and g to C meet the following conditions: 1. f C and gC are one-to-one. 2. The range of f C , the range of gC , and the set B are pairwise disjoint. The main result of this section, the recursion theorem, says that if C is freely generated then a function h on B always has an extension h on C that follows the sorts of rules considered above. RECURSION THEOREM Assume that the subset C of U is freely generated from B by f and g, where f : U × U → U, g : U → U. Further assume that V is a set and F, G, and h functions such that h : B → V, F : V × V → V, G : V → V. Then there is a unique function h:C →V such that (i) For x in B, h(x) = h(x); (ii) For x, y in C, h( f (x, y)) = F(h(x), h(y)), h(g(x)) = G(h(x)). Viewed algebraically, the conclusion of this theorem says that any map h of B into V can be extended to a homomorphism h from C (with operations f and g) into V (with operations F and G). If the content of the recursion theorem is not immediately apparent, try looking at it chromatically. You want to have a function h that paints each member of C some color. You have before you 1. h, telling you how to color the initial elements in B; 2. F, which tells you how to combine the color of x and y to obtain the color of f (x, y) (i.e., it gives h( f (x, y)) in terms of h(x) and h(y));

40

A Mathematical Introduction to Logic 3. G, which similarly tells you how to convert the color of x into the color of g(x). The danger is that of a conﬂict in which, for example, F is saying “green” but G is saying “red” for the same point (unlucky enough to be equal both to f (x, y) and g(z) for some x, y, z). But if C is freely generated, then this danger is avoided. EXAMPLES.

Consider again the examples of the preceding subsection.

1. B = {0} with one operation, the successor operation S. Then C is the set N of natural numbers. Since the successor operation is one-to-one and 0 is not in its range, C is freely generated from {0} by S. Therefore, by the recursion theorem, for any set V , any a ∈ V , and any F : V → V there is a unique h : N → V such that h(0) = a and h(S(x)) = F(h(x)) for each x ∈ N. For example, there is a unique h : N → N such that h(0) = 0 and h(S(x)) = 1 − h(x). This function has the value 0 at even numbers and the value 1 at odd numbers. 2. The integers are generated from {0} by the successor and predecessor operations but not freely generated. 3. Freeness fails also for the generation of the algebraic functions in the manner described. 4. The wffs are freely generated from the sentence symbols by the ﬁve formula-building operations. This fact is implicit in the parsing algorithm of the preceding section; we now want to focus on it here: UNIQUE READABILITY THEOREM The ﬁve formula-building operations, when restricted to the set of wffs, (a) Have ranges that are disjoint from each other and from the set of sentence symbols, and (b) Are one-to-one. In other words, the set of wffs is freely generated from the set of sentence symbols by the ﬁve operations. PROOF. that

To show that the restriction of E∧ is one-to-one, suppose (α ∧ β) = (γ ∧ δ),

where α, β, γ , and δ are wffs. Delete the ﬁrst symbol of each sequence, obtaining α ∧ β) = γ ∧ δ). Then we must have α = γ , lest one be a proper initial segment of the other (in contradiction to Lemma 13B). And then it follows at

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once that β = δ. The same argument applies to E∨ , E → , and E↔ ; for E¬ a simpler argument sufﬁces. A similar line of reasoning tells us that the operations have disjoint ranges. For example, if (α ∧ β) = (γ → δ) where α, β, γ , and δ are wffs, then as in the above paragraph we have α = γ . But that implies that ∧ = →, contradicting the fact that our symbols are distinct. Hence E∧ and E → (when restricted to wffs) have disjoint ranges. Similarly for any two binary connectives. The remaining cases are simple. If (¬ α) = (β ∧ γ ), then β begins with ¬, which no wff does. No sentence symbol is a sequence of symbols beginning with (. Now let us return to the question of extending a truth assignment v to v. First consider the special case where v is a truth assignment for the set of all sentence symbols. Then by applying the unique readability theorem and the recursion theorem we conclude that there is a unique extension v to the set of all wffs with the desired properties. Next take the general case where v is a truth assignment for a set S of sentence symbols. The set S generated from S by the ﬁve formulabuilding operations is freely generated, as a consequence of the unique readability theorem. So by the recursion theorem there is a unique extension v of v to that set, having the desired properties. EXAMPLE. We can apply the recursion theorem to establish that there is a unique function h deﬁned on the set of wffs such that h(A) = 1 for a sentence symbol h((¬ α)) = 3 + h(α), h((α ∧ β)) = 3 + h(α) + h(β),

A,

and similarly for ∨, →, and ↔. This function gives the length of each wff. PROOF OF THE RECURSION THEOREM. The idea is to let h be the union of many approximating functions. Temporarily call a function v (which maps part of C into V ) acceptable if it meets the conditions imposed on h by (i) and (ii). More precisely, v is acceptable iff the domain of v is a subset of C, the range a subset of V, and for any x and y in C: (i ) If x belongs to B and to the domain of v, then v(x) = h(x). (ii ) If f (x, y) belongs to the domain of v, then so do x and y, and v( f (x, y)) = F(v(x), v(y)). If g(x) belongs to the domain of v, then so does x, and v(g(x)) = G(v(x)).

42

A Mathematical Introduction to Logic Let K be the collection of all acceptable functions, and let h = K , the union of all the acceptable functions. Thus x, z ∈ h

iff x, z belongs to some acceptable v iff v(x) = z for some acceptable v.

(1.1)

We claim that h meets our requirements. The argument is settheoretic, and comprises four steps. First, here is an outline of four steps: 1. We claim that h is a function (i.e., that it is single-valued). Let S = {x ∈ C | for at most one z, x, z ∈ h} = {x ∈ C | all acceptable functions deﬁned at x agree there} It is easy to verify that S is inductive, by using (i ) and (ii ). Hence S = C and h is a function. 2. We claim that h ∈ K ; i.e., that h itself is an acceptable function. This follows fairly easily from the deﬁnition of h and the fact that it is a function. 3. We claim that h is deﬁned throughout C. It sufﬁces to show that the domain of h is inductive. It is here that the assumption of freeness is used. For example, one case is the following: Suppose that x is in the domain of h. Then h; g(x), G(h(x)) is acceptable. (The freeness is required in showing that it is acceptable.) Consequently, g(x) is in the domain of h. 4. We claim that h is unique. For given two such functions, let S be the set on which they agree. Then S is inductive, and so equals C. Now for the details. 1. As above, let S = {x ∈ C | for at most one z, x, z ∈ h} = {x ∈ C | all acceptable functions deﬁned at x agree there} Toward showing that S is inductive, ﬁrst consider some x in B. Suppose that v1 and v2 are acceptable functions deﬁned at x; we seek to show that v1 (x) = v2 (x). But condition (i ) tells us that both v1 (x) and v2 (x) must equal h(x), so indeed v1 (x) = v2 (x). This shows that x ∈ S; since x was an arbitrary member of B we have B ⊆ S. Secondly we must check that S is closed under f and g. So suppose that some x and y are in S; we ask whether f (x, y) is in S. So suppose that v1 and v2 are acceptable functions deﬁned at

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f (x, y); we seek to show that they agree there. But condition (ii ) tells us that v1 ( f (x, y)) = F(v1 (x), v1 (y)) and v2 ( f (x, y)) = F(v2 (x), v2 (y)). And because x and y are in S, we have v1 (x) = v2 (x) and v1 (y) = v2 (y) (and these are deﬁned). So we conclude that v1 ( f (x, y)) = v2 ( f (x, y)). This shows that f (x, y) ∈ S. Hence S is closed under f . A similar argument shows that S is closed under g. Thus S is inductive and so S = C. This shows that h is singlevalued, i.e., is a function. Because h includes every acceptable function as a subset, we can say that h(x) = v(x) whenever v is an acceptable function and x ∈ dom v. 2. We claim that h is acceptable. Clearly dom h ⊆ C and ran h ⊆ V (by (∗)), and we have just veriﬁed that h is a function. It remains to check that h satisﬁes conditions (i ) and (ii ). First we examine (i ). Assume x ∈ B and x ∈ dom h (so that x, h(x) ∈ h). There must be some acceptable v with v(x) = h(x). Because v satisﬁes (i ), we have v(x) = h(x) whence h(x) = h(x). So h satisﬁes (i ). Secondly we examine (ii ). Assume that f (x, y) ∈ dom h. Again there must be some acceptable v with v( f (x, y)) = h( f (x, y)). Because v satisﬁes (ii ), we have v( f (x, y)) = F(v(x), v(y)). Now h(x) = v(x) and h(y) = v(y) and hence h( f (x, y)) = v( f (x, y)) = F(v(x), v(y)) = F(h(x), h(y)). In a similar way, we ﬁnd that h(g(x)) = G(h(x)) whenever g(x) ∈ dom h. Hence h meets condition (ii ) and so is acceptable. 3. Next we must show that dom h is inductive. First consider a point x in B. Then the set { x, h(x) } is a (small) acceptable function. For it clearly satisﬁes (i ). It also satisﬁes (ii ) because / ran gC . Thus { x, h(x) } is acceptable and x ∈ / ran f C and x ∈ therefore is included in h. Hence x ∈ dom h. This shows that B ⊆ dom h. We further claim that dom h is closed under f and g. Toward this end, consider any s and t in dom h. We hope that f (s, t) ∈ dom h. But if not, then let v = h ∪ { f (s, t), F(h(s), h(t)) }, the result of adding this one additional pair to h. It is clear that v is a function, dom v ⊆ C, and ran v ⊆ V . We claim that v satisﬁes (i ) and (ii ).

44

A Mathematical Introduction to Logic First take (i ). If x ∈ B ∩ dom v then x = f (s, t), by freeness. Hence x ∈ dom h and we have v(x) = h(x) = h(x). Next take (ii ). Assume that f (x, y) ∈ dom v for some x and y in C. If f (x, y) ∈ dom h then v( f (x, y)) = h( f (x, y)) = F(h(x), h(y)) = F(v(x), v(y)) since h is acceptable. The other possibility is that f (x, y) = f (s, t). Then by freeness we have x = s and y = t, and we know that these points are in dom h ⊆ dom v. By construction, v( f (s, t)) = F(h(s), h(t)) = F(v(s), v(t)). Finally suppose that g(x) ∈ dom v for x in C. Then by freeness we have g(x) = f (s, t). Hence g(x) ∈ dom h, and consequently v(g(x)) = h(g(x)) = G(h(x)) = G(v(x)). Thus v is acceptable. But that tells us that v ⊆ h, so that f (s, t) ∈ dom h after all. A similar argument shows that dom h is closed under g as well. Hence dom h is inductive and therefore coincides with C. 4. To show that h is unique, suppose that h 1 and h 2 both satisfy the conclusion of the theorem. Let S be the set on which they agree: S = {x ∈ C | h 1 (x) = h 2 (x)}. Then it is not hard to verify that S is inductive. Consequently S = C and h 1 = h 2 . One ﬁnal comment on induction and recursion: The induction principle we have stated is not the only one possible. It is entirely possible to give proofs by induction (and deﬁnitions by recursion) on the length of expressions, the number of places at which connective symbols occur, etc. Such methods are inherently less basic but may be necessary in some situations.

Exercises 1. Suppose that C is generated from a set B = {a, b} by the binary operation f and unary operation g. List all the members of C2 . How many members might C3 have? C4 ? 2. Obviously (A3 → ∧A4 ) is not a wff. But prove that it is not a wff. 3. We can generalize the discussion in this section by requiring of F only that it be a class of relations on U . C is deﬁned as before, except that x0 , x1 , . . . , xn is now a construction sequence provided that for each i ≤ n we have either xi ∈ B or x j1 , . . . , x jk , xi ∈ R for some R ∈ F and some j1 , . . . , jk all less than i. Give the correct deﬁnition of C and show that C = C .

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SECTION 1.5 Sentential Connectives We have thus far employed ﬁve sentential connective symbols. Even in the absence of a general deﬁnition of “connective,” it is clear that the ﬁve familiar ones are not the only ones possible. Would we gain anything by adding more connectives to the language? Would we lose anything by omitting some we already have? In this section we make these questions precise and give some answers. First consider an informal example. We could expand the language by adding a three-place sentential connective symbol #, called the majority symbol. We allow now as a wff the expression (#αβγ ) whenever α, β, and γ are wffs. In other words, we add a sixth formulabuilding operator to our list: E# (α, β, γ ) = (#αβγ ). Then we must give the interpretation of this symbol. That is, we must say how to compute v((#αβγ )), given the values v(α), v(β), and v(γ ). We choose to deﬁne v((#αβγ )) is to agree with the majority of v(α), v(β), v(γ ). We claim that this extension has gained us nothing, in the following precise sense: For any wff in the extended language, there is a tautologically equivalent wff in the original language. (On the other hand, the wff in the original language may be much longer than the wff in the extended language.) We will prove this (in a more general situation) below; here we just note that it relies on the fact that (#αβγ ) is tautologically equivalent to (α ∧ β) ∨ (α ∧ γ ) ∨ (β ∧ γ ). (We note parenthetically that our insistence that v((#αβγ )) be calculable from v(α), v(β), v(γ ) plays a deﬁnite role here. In everyday speech, there are unary operators like “it is possible that” or “I believe that.” We can apply one of these operators to a sentence, producing a new sentence whose truth or falsity cannot be determined solely on the basis of the truth or falsity of the original one.) In generalizing the foregoing example, the formal language will be more of a hindrance than a help. We can restate everything using only functions. Say that a k-place Boolean function is a function from {F, T }k into {F, T }. (A Boolean function is then anything which is a k-place Boolean function for some k. We stretch this slightly by permitting F and T themselves to be 0-place Boolean functions.) Some sample

46

A Mathematical Introduction to Logic Boolean functions are deﬁned by the equations (where X ∈ {F, T }) Iin (X 1 , . . . , X n ) = X i , N (F) = T, N (T ) = F, K (T, T ) = T, K (F, X ) = K (X, F) = F, A(F, F) = F, A(T, X ) = A(X, T ) = T, C(T, F) = F, C(F, X ) = C(X, T ) = T, E(X, X ) = T, E(T, F) = E(F, T ) = F. From a wff α we can extract a Boolean function. For example, if α is the wff A1 ∧ A2 , then we can make a table, Table V. The 22 lines of the table correspond to the 22 truth assignments for {A1 , A2 }. For each of the 22 pairs X , we set Bα ( X ) equal to the truth value α receives when its sentence symbols are given the values indicated by X . TABLE V A1

A2

A1 ∧ A2

F F T T

F T F T

F F F T

Bα (F, F) = F Bα (F, T ) = F Bα (T, F) = F Bα (T, T ) = T

In general, suppose that α is a wff whose sentence symbols are at most A1 , . . . , An . We deﬁne an n-place Boolean function Bαn (or just Bα if n seems unnecessary), the Boolean function realized by α, by Bαn (X 1 , . . . , X n ) = the truth value given to α when A1 , . . . , An are given the values

X 1, . . . , X n .

Bαn (X 1 , . . . ,

Or, in other words, X n ) = v(α), where v is the truth assignment for {A1 , . . . , An } for which v(Ai ) = X i . Thus Bαn comes from looking at v(α) as a function of v, with α ﬁxed. For example, the Boolean functions listed previously are obtainable in this way: Iin = BAn i N = B¬1 A1 , K = BA2 1 ∧A2 , A = BA2 1 ∨A2 , C = BA2 1 →A2 , E = BA2 1 ↔A2 . From these functions we can compose others. For example, B¬2 A1 ∨¬ A2 (X 1 , X 2 ) = A(N (I12 (X 1 , X 2 )), N (I22 (X 1 , X 2 ))).

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(The right-hand side of this equation can be compared with the result of putting ¬ A1 ∨ ¬ A2 into Polish notation.) We will shortly come to the question whether every Boolean function is obtainable in this fashion. As the theorem below states, in shifting attention from wffs to the Boolean functions they realize, we have in effect identiﬁed tautologically equivalent wffs. Impose an ordering on {F, T } by deﬁning F < T . (If F = 0 and T = 1, then this is the natural order.) THEOREM 15A Let α and β be wffs whose sentence symbols are among A1 , . . . , An . Then (a) α |= β iff for all X ∈ {F, T }n , Bα ( X ) ≤ Bβ ( X ). (b) α |==| β iff Bα = Bβ . (c) |= α iff Bα is the constant function with value T . PROOF OF (A). α |= β iff for all 2n truth assignments v for A1 , . . . , An , whenever v(α) = T , then also v(β) = T . (This is true even if the sentence symbols in α and β do not include all of A1 , . . . , An ; cf. Exercise 6 of Section 1.2.) Thus α |= β iff for all 2n assignments v, v(α) = T ⇒ v(β) = T, iff for all 2n n-tuples X , Bαn ( X ) = T ⇒ Bβn ( X ) = T, n Bαn ( X ) ≤ Bβn ( X ), iff for all 2 n-tuples X , where F < T .

In addition to identifying tautologically equivalent wffs, we have freed ourselves from the formal language. We are now at liberty to consider any Boolean function, whether it is realized by a wff or not. But this freedom is only apparent: THEOREM 15B Let G be an n-place Boolean function, n ≥ 1. We can ﬁnd a wff α such that G = Bαn , i.e., such that α realizes the function G. The primary reason for introducing Boolean functions is to allow us to formulate this theorem. The theorem was stated by Emil Post in 1921. PROOF. Case I: G is the constant function with value F. Let α = A1 ∧ ¬ A1 . Case II: Otherwise there are k points at which G has the value T , k > 0. List these: X 1 = X 11 , X 12 , . . . , X 1n , X 2 = X 21 , X 22 , . . . , X 2n , ··· X k = X k1 , X k2 , . . . , X kn .

48

A Mathematical Introduction to Logic Let

Aj iff X i j = T, (¬ A j ) iff X i j = F, γi = βi1 ∧ · · · ∧ βin , α = γi ∨ γ2 ∨ · · · ∨ γk .

βi j =

We claim that G = Bαn . At this point it might be helpful to consider a concrete example. Let G be the three-place Boolean function as follows: G(F, F, F) = F, G(F, F, T ) = T, G(F, T, F) = T, G(F, T, T ) = F, G(T, F, F) = T, G(T, F, T ) = F, G(T, T, F) = F, G(T, T, T ) = T. Then the list of triples at which G assumes the value T has four members: F F T ¬ A1 ∧ ¬ A2 ∧ A3 , F T F ¬ A1 ∧ A2 ∧ ¬ A3 , T FF A1 ∧ ¬ A2 ∧ ¬ A3 , TTT A1 ∧ A2 ∧ A3 . To the right of each triple above is written the corresponding conjunction γi . Then α is the formula (¬ A1 ∧ ¬ A2 ∧ A3 ) ∨ (¬ A1 ∧ A2 ∧ ¬ A3 ) ∨ (A1 ∧ ¬ A2 ∧ ¬ A3 ) ∨ (A1 ∧ A2 ∧ A3 ). Notice how α lists explicitly the triples at which G assumes the value T . To return to the proof of the theorem, note ﬁrst that Bαn ( X i ) = T for 1 ≤ i ≤ k. (For the truth assignment corresponding to X i satisﬁes γi and hence satisﬁes α.) On the other hand, only one truth assignment for {A1 , . . . , An } can satisfy γi , whence only k such truth assignments can satisfy α. Hence Bαn ( X ) = F for the 2n − k other n-tuples Y . Thus in all cases, Bαn (Y ) = G(Y ). From this theorem we know that every Boolean function is realizable. Of course the α that realizes G is not unique; any tautologically equivalent wff will also realize the same function. It is sometimes of interest to choose α to be as short as possible. (In the example done above, the wff A1 ↔ A2 ↔ A3 also realizes G.)

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As a corollary to the above theorem, we may conclude that we have enough (in fact, more than enough) sentential connectives. For suppose that we expand the language by adding some exotic new sentential connectives (such as the majority connective discussed at the beginning of this section). Any wff ϕ of this expanded language realizes a Boolean function Bϕn . By the above theorem we have a wff α of the original language such that Bϕn = Bαn . Hence ϕ and α are tautologically equivalent, by Theorem 15A. In fact, the proof shows that α can be of a rather special form. For one thing, the only sentential connective symbols in α are ∧, ∨, and ¬. Furthermore, α is in so-called disjunctive normal form (abbreviated DNF). That is, α is a disjunction α = γ1 ∨ · · · ∨ γk , where each γi is a conjunction γi = βi1 ∧ · · · ∧ βini and each βi j is a sentence symbol or the negation of a sentence symbol. (The advantages of wffs in disjunctive normal form stem from the fact that they explicitly list the truth assignments satisfying the formula.) Thus we have the consequence: COROLLARY 15C For any wff ϕ, we can ﬁnd a tautologically equivalent wff α in disjunctive normal form. Because every function G : {F, T }n → {F, T } for n ≥ 1 can be realized by a wff using only the connective symbols in {∧, ∨, ¬}, we say that the set {∧, ∨, ¬} is complete. (Actually the completeness is more a property of the Boolean functions K , A, and N that correspond to these symbols. But the above terminology is convenient.) Once we have a complete set of connectives, we know that any wff is tautologically equivalent to one all of whose connectives are in that set. (For given any wff ϕ, we can make α using those connectives and realizing Bϕ . Then α |==| ϕ.) The completeness of {∧, ∨, ¬} can be improved upon: THEOREM 15D

Both {¬, ∧} and {¬, ∨} are complete.

PROOF. We must show that any Boolean function G can be realized by a wff using only, in the ﬁrst case, {¬, ∧}. We begin with a wff α using {¬, ∧, ∨} that realizes G. It sufﬁces to ﬁnd a tautologically equivalent α that uses only {¬, ∧}. For this we use De Morgan’s law: β ∨ γ |==| ¬ (¬ β ∧ ¬ γ ). By applying this repeatedly, we can completely eliminate ∨ from α.

50

A Mathematical Introduction to Logic (More formally, we can prove by induction on α that there is a tautologically equivalent α in which only the connectives ∧, ¬ occur. Two cases in the inductive step are Case ¬: If α is (¬ β), then let α be (¬ β ). Case ∨: If α is (β ∨ γ ), then let α be ¬ (¬ β ∧ ¬ γ ). Since β and γ are tautologically equivalent to β and γ , respectively, α = |==| |==| =

¬ (¬ β ∧ ¬ γ ) ¬ (¬ β ∧ ¬ γ ) β ∨γ α.

In future proofs that a set of connectives is complete, this induction will be omitted. Instead we will just give, for example, the method of simulating ∨ by using ¬ and ∧. Showing that a certain set of connectives is not complete is usually more difﬁcult than showing that one is complete. The basic method is ﬁrst to show (usually by induction) that for any wff α using only those connectives, the function Bαn has some peculiarity, and secondly to show that some Boolean function lacks that peculiarity. EXAMPLE.

{∧, →} is not complete.

PROOF. The idea is that with these connectives, if the sentence symbols are assigned T , then the entire formula is assigned T . In particular, there is nothing tautologically equivalent to ¬ A. In more detail, we can show by induction that for any wff α using only these connectives and having A as its only sentence symbol, we have A |= α. (In terms of functions, this says that Bα1 (T ) = T .) The same argument shows that {∧, ∨, →, ↔} is not complete. n

For each n there are 22 n-place Boolean functions. Hence if we identify a connective with its Boolean function (e.g., ∧ with the function K n mentioned before), we have 22 n-ary connectives. We will now catalog these for n ≤ 2.

0-ary Connectives There are two 0-place Boolean functions, F and T . For the corresponding connective symbols we take ⊥ and . Now an n-ary connective symbol combines with n wffs to produce a new wff. When n = 0, we have that ⊥ is a wff all by itself. It differs from the sentence symbols in that v(⊥) = F for every v; i.e., ⊥ is a logical symbol always assigned the value F. Similarly, is a wff, and v() = T for every v. Then, for example, A → ⊥ is a wff, tautologically equivalent to ¬ A, as can be seen from a two-line truth table.

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Unary Connectives There are four unary connectives but only one of any interest. The interesting case is negation. The other three one-place Boolean functions are the identity function and the two constant functions.

Binary Connectives There are 16 binary connectives, but only the last 10 listed in Table VI are “really binary.” TABLE VI Symbol

Equivalent

Remarks

∧

⊥ A B ¬A ¬B A∧B

two-place constant, essentially 0-ary two-place constant, essentially 0-ary projection, essentially unary projection, essentially unary negation, essentially unary negation, essentially unary and; if F = 0 and T = 1, then this gives multiplication in the ﬁeld {0, 1} or (inclusive) conditional biconditional reversed conditional exclusive or, “A or B and not both”; if F = 0 and T = 1, then this gives the usual addition (modulo 2) in the ﬁeld {0, 1} nor, “neither A nor B” nand, “not both A and B”; the symbol is called the Sheffer stroke the usual ordering, where F < T the usual ordering, where F < T

∨ → ↔ ←

+

A∨B A→B A↔B A←B (A ∨ B) ∧ ¬ (A ∧ B)

↓ |

¬ (A ∨ B) ¬ (A ∧ B)

< >

(¬ A) ∧ B A ∧ (¬ B)

Ternary Connectives There are 256 ternary connectives; 2 are essentially 0-ary, 6 (= 2 · ( 31 )) are essentially unary, and 30 (= 10 · ( 32 )) are essentially binary. This leaves 218 that are really ternary. We have thus far mentioned only the majority connective #. There is, similarly, the minority connective. In Exercise 7 we encounter +3 , ternary addition modulo 2. +3 αβγ is assigned the value T iff an odd number of α, β, and γ are assigned T . This formula is equivalent both to α + β + γ and to α ↔ β ↔ γ . Another ternary connective arises in Exercise 8.

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A Mathematical Introduction to Logic EXAMPLE. PROOF.

{|} and {↓} are complete. For | ¬ α |==| α | α α ∨ β |==| (¬ α) | (¬ β).

Since {¬, ∨} is complete and ¬, ∨ can be simulated using only |, we conclude that {|} is complete. EXAMPLE. {¬, →} is complete. In fact, of the 10 connectives that are really binary, eight have the property of forming, when added to ¬, a complete set. The two exceptions are + and ↔; see Exercise 5. EXAMPLE. {⊥, →} is complete. In fact, because with this set we can realize even the two 0-place Boolean functions, it is supercomplete.

Exercises 1. Let G be the following three-place Boolean function. G(F, F, F) = T, G(F, F, T ) = T, G(F, T, F) = T, G(F, T, T ) = F,

2. 3. 4.

5.

G(T, F, F) = T, G(T, F, T ) = F, G(T, T, F) = F, G(T, T, T ) = F.

(a) Find a wff, using at most the connectives ∨, ∧, and ¬, that realizes G. (b) Then ﬁnd such a wff in which connective symbols occur at not more than ﬁve places. Show that | and ↓ are the only binary connectives that are complete by themselves. Show that {¬, #} is not complete. Let M be the ternary minority connective. (Thus v(Mαβγ ) always disagrees with the majority of v(α), v(β), and v(γ ).) Show the following: (a) {M, ⊥} is complete. (b) {M} is not complete. Show that {, ⊥, ¬, ↔, +} is not complete. Suggestion: Show that any wff α using these connectives and the sentence symbols A and B has an even number of T ’s among the four possible values of v(α). Remark: Another approach uses the algebra of the ﬁeld {0, 1}. Any Boolean function realizable with these connectives has a certain linearity property.

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6. Show that {∧, ↔, +} is complete but that no proper subset is complete. 7. Let +3 be the ternary connective such that +3 αβγ is equivalent to α + β + γ. (a) Show that {, ⊥, ∧, +3 } is complete. (b) Show that no proper subset is complete. Remark: +3 is the ternary parity connective; the condition for v(+α1 α2 α3 ) = T is that v(αi ) = T for an odd number of i’s. + is the binary parity connective. The function G in the proof of Theorem 15B is the 3-place parity function. 8. Let I be the ternary connective such that Iαβγ is assigned the value T iff exactly one of the formulas α, β, γ is assigned the value T . Show that there are no binary connectives ◦ and such that Iαβγ is equivalent to (α ◦ β)γ . 9. Say that a formula α is in conjunctive normal form (abbreviated CNF) iff it is a conjunction α = γ1 ∧ · · · ∧ γk where each γi is a disjunction γi = βi1 ∨ · · · ∨ βin and each βi j is either a sentence symbol or the negation of a sentence symbol. (a) Find a formula in conjunctive normal form that is tautologically equivalent to A ↔ B ↔ C. (b) Show that for any formula, we can ﬁnd a tautologically equivalent formula in conjunctive normal form. 10. Add the 0-place connectives , ⊥ to our language. For each wff ϕ and sentence symbol A, let ϕA be the wff obtained from ϕ by replacing A by . Similarly for ϕ⊥A . Then let ϕ∗A = (ϕA ∨ ϕ⊥A ). Prove the following: (a) ϕ |= ϕ∗A . (b) If ϕ |= ψ and A does not appear in ψ, then ϕ∗A |= ψ. (c) The formula ϕ is satisﬁable iff ϕ∗A is satisﬁable. Remarks: We can think of ϕ∗A as trying to say everything ϕ says, but without being able to use the symbol A. Parts (a) and (b) state that ϕ∗A is the strongest A-free consequence of ϕ. The formulas ϕ and ϕ∗A are not tautologically equivalent in general, but they are “equally satisﬁable” by part (c). The operation of forming ϕ∗A from ϕ is (in another context) called resolution on A. 11. (Interpolation theorem) If α |= β, then there is some γ all of whose sentence symbols occur both in α and in β and such that α |= γ |= β. Suggestion: Use the preceding exercise.

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A Mathematical Introduction to Logic Remarks: There is an analogue of Exercise 11 that holds for ﬁrstorder logic. But the proof in that case is very different, because there is no analogue of Exercise 10. 12. Is the set {∧, , ⊥} complete? Support your answer.

SECTION 1.6 Switching Circuits1 Consider an electrical device (traditionally a black box) having n inputs and one output (Fig. 1). Assume that to each input we apply a signal having one of two values and that the output has one of two values. The two possible values we call F and T . (We could also deﬁne the F value as 0 potential and choose the unit of potential so that the T value has potential 1.) Further assume that the device has no memory; i.e., the present output level depends only on the present inputs (and not on past history). Then the performance of the device is described by a Boolean function: G(X 1 , . . . , X n ) = the output level given the input signalsX 1 , . . . , X n .

Figure 1. Electrical device with three inputs.

Devices meeting all these assumptions constitute an essential part of digital-computer circuitry. There is, for example, the two-input AND gate, for which the output is the minimum of the inputs (where F < T ). This device realizes the Boolean function K of the preceding section. It is convenient to attach the labels A1 and A2 to the inputs and to label the output A1 ∧ A2 . Similar devices can be made for other sentential connectives. For a two-input OR gate (Fig. 2) the output voltage is the maximum of the input voltages. Corresponding to the negation connective there is the NOT device (or inverter), whose output voltage is the opposite of the input voltage. A circuit can be constructed from various devices of this sort. And it is again natural to use wffs of our formal language to label the voltages

1

This section, which discusses an application of the ideas of previous sections, may be omitted without loss of continuity.

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Figure 2. OR gate.

at different points (Fig. 3). Conversely, given the wff thus attached to the output, we can approximately reconstruct the circuit, which looks very much like the tree of the wff’s formation.

Figure 3. Circuit with wffs as labels.

For example, the circuit for ((A ∧ B) ∧ D) ∨ ((A ∧ B) ∧ ¬ C) would probably be as shown in Fig. 4. Duplication of the circuit for A ∧ B would not usually be desirable.

Figure 4. Circuit for ((A ∧ B) ∧ D) ∨ ((A ∧ B) ∧ ¬C).

Tautologically equivalent wffs yield circuits having ultimately the

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A Mathematical Introduction to Logic same performance, although possibly at different cost and (if the devices are not quite instantaneous in operation) different speed. Deﬁne the delay (also called the depth) of a circuit as the maximum number of boxes through which the signal can pass in going from an input to the output. The corresponding notion for formulas is conveniently deﬁned by recursion. 1. The delay of a sentence symbol is 0. 2. The delay of ¬ α is one greater than the delay of α. 3. The delay of α∧β is one greater than the maximum of the delay of α and the delay of β. And similarly for any other connective. For example, the circuit of (A1 ∧ A2 ) ∨ ¬ A3 uses three devices and has a delay of 2. The tautologically equivalent formula ¬ (A3 ∧ (¬ A1 ∨ ¬ A2 )) gives a circuit having ﬁve devices and a delay of 4. The problem facing many a computer engineer can be stated: Given a circuit (or its wff), ﬁnd an equivalent circuit (or a tautologically equivalent wff) for which the cost is a minimum, subject to constraints such as a maximum allowable delay. For this problem there is some catalog of available devices; for example, she might have available NOT, two-input AND, three-input OR. (It is clearly desirable that the available devices correspond to a complete set of connectives.) The catalog of devices determines a formal language, having a connective symbol for each device. EXAMPLE 1. Inputs: A, B, C. Output: To agree with the majority of A, B, and C. Devices available: two-input OR, two-input AND. One solution is ((A ∧ B) ∨ (A ∧ C)) ∨ (B ∧ C), which uses ﬁve devices and has a delay of 3. But a better solution is (A ∧ (B ∨ C)) ∨ (B ∧ C), which uses four devices and has the same delay. Furthermore, there is no solution using only three devices; cf. Exercise 1. EXAMPLE 2. Inputs: A and B. Output: T if the inputs agree, F otherwise; i.e., the circuit is to test for equality. Device available: two-input NOR. One solution is ((A ↓ A) ↓ B) ↓ ((B ↓ B) ↓ A). This uses ﬁve devices; is there a better solution? And there is

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the deeper question: Is there an efﬁcient procedure for ﬁnding a minimal solution? These are questions that we merely raise here. In recent years a great deal of work has gone into investigating questions of this type. EXAMPLE 3. (Relay switching) Inputs: A, ¬ A, B, ¬ B, . . . Devices: OR (any number of inputs), AND (any number of inputs). Cost: Devices are free, but each use of an input costs one unit. To test for equality of A and B we could use (A ∧ B) ∨ (¬ A ∧ ¬ B). The wiring diagram for the circuit is shown in Fig. 5. The circuit will pass current iff A and B are assigned the same value. (This formula, equivalent to A ↔ B, has the property that its truth value changes whenever the value of one argument changes. For this reason, the circuit is used, with double-throw switches, in wiring hallway lights.)

Figure 5. Wiring diagram for (A ∧ B) ∨ (¬A ∧ ¬B).

But there is one respect in which relay circuits do not ﬁt the description given at the beginning of this section. Relays are bilateral devices; they will pass current in either direction. This feature makes “bridge” circuits possible (Fig. 6). The methods described here do not apply to such circuits.

Figure 6. Bridge circuit.

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A Mathematical Introduction to Logic EXAMPLE 4. There are four inputs, and the circuit is to realize the Boolean function G, where G is to have the value T at F, F, F, T , F, F, T, F , F, F, T, T , F, T, F, F , F, T, F, T , F, T, T, F , F, T, T, T , and T, F, F, T . G is to have the value F at T, F, F, F , T, F, T, F , T, T, F, F , T, T, T, F , and T, T, T, T . At the remaining three points, F, F, F, F , T, F, T, T , and T, T, F, T , we don’t care about the value of G. (The application of the circuit is such that these three combinations never occur.) We know that G can be realized by using, say, { ∧ , ∨, ¬}. But we want to do this in an efﬁcient way. The ﬁrst step is to represent the data in a more comprehensible form. We can do this by means of Fig. 7. Since G(F, F, F, T ) = T , we have placed a T in the square with coordinates ¬ A, ¬ B, ¬ C, D . Similarly, there is an F in the square with coordinates A, B, ¬ C, ¬ D because G(T, T, F, F) = F. The three squares we do not care about are left blank.

Figure 7. Diagram for Example 4.

Now we look for a simple geometrical pattern. The shaded area includes all T ’s and no F’s. It corresponds to the formula ( ¬ A) ∨ ( ¬ C ∧ D), which is reasonably simple and meets all our requirements. Note that the input B is not needed at all.

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Exercises 1. In Example 1 of this section, verify that there is no solution using only three devices. 2. Deﬁne a literal to be a wff which is either a sentence symbol or the negation of a sentence symbol. An implicant of ϕ is a conjunction α of literals (using distinct sentence symbols) such that α |= ϕ. We showed in Section 1.5 (cf. Corollary 15C) that any satisﬁable wff ϕ is tautologically equivalent to a disjunction α1 ∨ · · · ∨αn where each αi is an implicant of ϕ. An implicant α of ϕ is prime iff it ceases to be an implicant upon the deletion of any of its literals. Any disjunction of implicants equivalent to ϕ clearly must, if it is to be of minimum length, consist only of prime implicants. (a) Find all prime implicants of (A → B) ∧ ( ¬ A → C). (b) Which disjunctions of prime implicants enjoy the property of being tautologically equivalent to the formula in part (a)? 3. Repeat (a) and (b) of Exercise 2, but for the formula (A ∨ ¬ B) ∧ ( ¬ C ∨ D) → B ∧ ((A ∧ C) ∨ ( ¬ C ∧ D)).

SECTION 1.7 Compactness and Effectiveness Compactness We now give a proof of the compactness theorem mentioned earlier (Section 1.2). Call a set of wffs satisﬁable iff there is a truth assignment that satisﬁes every member of . COMPACTNESS THEOREM A set of wffs is satisﬁable iff every ﬁnite subset is satisﬁable. Let us temporarily say that is ﬁnitely satisﬁable iff every ﬁnite subset of is satisﬁable. Then the compactness theorem asserts that this notion coincides with satisﬁability. Notice that if is satisﬁable, then automatically it is ﬁnitely satisﬁable. Also if is ﬁnite, then the converse is trivial. (Every set is a subset of itself.) The nontrivial part is to show that if an inﬁnite set is ﬁnitely satisﬁable, then it is satisﬁable. PROOF OF THE COMPACTNESS THEOREM. The proof consists of two distinct parts. In the ﬁrst part we take our given ﬁnitely satisﬁable set and extend it to a maximal such set . In the second part we utilize to make a truth assignment that satisﬁes .

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A Mathematical Introduction to Logic For the ﬁrst part let α1 , α2 , . . . be a ﬁxed enumeration of the wffs. (This is possible since the set of sentence symbols, and hence the set of expressions, is countable; see Theorem 0B.) Deﬁne by recursion (on the natural numbers) 0 = , n ; αn+1 n+1 = n ; ¬ αn+1

if this is ﬁnitely satisﬁable, otherwise.

(Recall that n ; αn+1 = n ∪ {αn+1}.) Then each n is ﬁnitely satisﬁable; see Exercise 1. Let = n n , the limit of the n ’s. It is clear that (1) ⊆ and that (2) for any wff α either α ∈ or (¬ α) ∈ . Furthermore, (3) is ﬁnitely satisﬁable. For any ﬁnite subset is already a ﬁnite subset of some n and hence is satisﬁable. This concludes the ﬁrst part of the proof; we now have a set having properties (1)–(3). There is in general not a unique such set, but there is at least one. (An alternative proof of the existence of such a — and one that we can use even if there are uncountably many sentence symbols — employs Zorn’s lemma. The reader familiar with uses of Zorn’s lemma should perceive its applicability here.) For the second part of the proof we deﬁne a truth assignment v for the set of all sentence symbols: v(A) = T

iff

A∈

for any sentence symbol A. Then for any wff ϕ, we claim that v satisﬁes ϕ

iff ϕ ∈ .

This is proved by induction on ϕ; see Exercise 2. Since ⊆ , v must then satisfy every member of . COROLLARY 17A 0 |= τ . PROOF. 0 |= τ

If |= τ , then there is a ﬁnite 0 ⊆ such that

We use the basic fact that |= τ iff ; ¬ τ is unsatisﬁable. for every ﬁnite 0 ⊆ ⇒ 0 ; ¬ τ is satisﬁable for every ﬁnite ⇒ ; ¬ τ is ﬁnitely satisﬁable ⇒ ; ¬ τ is satisﬁable ⇒ |= τ.

0 ⊆

In fact, the above corollary is equivalent to the compactness theorem; see Exercise 3.

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Effectiveness and Computability Although the method of truth tables is rather cumbersome to use, the existence of the method yields interesting theoretical conclusions. Suppose we ask of a set of wffs whether or not there is an effective procedure that, given a wff τ , will decide whether or not |= τ . By an effective procedure we mean one meeting the following conditions: 1. There must be exact instructions (i.e., a program) explaining how to execute the procedure. The instructions must be ﬁnite in length. After all, it must be possible to give the instructions to the person or machine doing the calculations, and one cannot give someone all of an inﬁnite object. But we impose no upper limit in advance on exactly how long the instructions can be. If there are more lines in the instructions than there are electrons in the universe, we merely shrug and say, “That’s a pretty long program.” 2. These instructions must not demand any brilliance or originality on the part of the person or machine following them. The idea is that a diligent clerk (who knows no mathematics but is very good at following directions) or your computing machine (which does not think at all) should be able to execute the instructions. That is, it must be possible for the instructions to be mechanically implemented. The procedure must avoid random devices (such as the ﬂipping of a coin), or any such device that can, in practice, only be approximated. 3. In the case of a decision procedure, as asked for above, the procedure must be such that, given a wff τ , after a ﬁnite number of steps the procedure produces a “yes” or “no” answer. (That is, the procedure must be an algorithm for determining the answer.) On the other hand, we place no bound in advance on the amount of time that might be required before the answer appears. Nor do we place any advance bound on the amount of scratch paper (or other storage medium) that might be required. These will depend on, among other things, the input τ . But for any one τ , the procedure is to require only a ﬁnite number of steps to produce the answer, and so only a ﬁnite amount of scratch paper will be consumed. It is no fair doing an inﬁnite number of steps and then giving the answer. People with a digital computing machine may regard a procedure as being effective only when it can be carried out on their machine in a “reasonable” length of time. Of course, the matter of what is reasonable may change with the circumstances. Next year they plan to get a faster machine with more memory. At that time, their idea of what can be done in a reasonable length of time will be considerably extended. We want here a concept of effective procedure that is the limiting case where all the practical restrictions on running time and memory space are removed.

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A Mathematical Introduction to Logic Of course the foregoing description can hardly be considered a precise deﬁnition of the word “effective.” And, in fact, that word will be used only in an informal intuitive way throughout this book. (In Chapter 3 we will meet a precise counterpart, “recursive.”) But as long as we restrict ourselves to positive assertions that there does exist an effective procedure of a certain sort, the informal approach sufﬁces. We simply display the procedure, show that it works, and people will agree that it is effective. (But this relies on the empirical fact that procedures which appear effective to one mathematician also appear so to others.) If we wanted a negative result, that there did not exist an effective procedure of a certain sort, then this informal viewpoint would be inadequate. (In Chapter 3 we do want to obtain just such negative results.) Because the notion of effectiveness is informal, deﬁnitions and theorems involving it will be marked with a star. For example:

THEOREM 17B There is an effective procedure that, given any expression ε, will decide whether or not it is a wff.

PROOF.

See the algorithm in Section 1.3 and the footnotes thereto.

There is one technical point here, stemming from the fact that our language has inﬁnitely many different sentence symbols. When we speak of being “given” an expression ε, we imagine that someone might write down the symbols in ε, one after the other. It is implausible that anyone has the potential of writing down any one of an inﬁnitude of symbols. To avoid this, we adopt the following “input/output format”: Instead of writing down A5 , for example, we use A , a string of ﬁve symbols. Now the total number of symbols in our alphabet is only nine: (, ), ¬, ∧, ∨, →, ↔, A, and . (If we identify these nine symbols with the digits 1–9, we get expressions that look particularly familiar in computational settings! And we still have the 0 digit for separating expressions.) Theorem 17B states that the set of wffs is decidable in the sense of the following deﬁnition:

DEFINITION. A set of expressions is decidable iff there exists an effective procedure that, given an expression α, will decide whether or not α ∈ .

For example, any ﬁnite set is decidable. (The instructions can simply list the ﬁnitely many members of the set. Then the algorithm can check the input against the list.) Some inﬁnite sets are decidable but not all. On the one hand, there are uncountably many (2ℵ0 , to be exact) sets of expressions. On the other hand, there can be only countably many effective procedures. This is because the procedure is completely deter-

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mined by its (ﬁnite) instructions. There are only ℵ0 ﬁnite sequences of letters, and the instructions, when written out, form a ﬁnite sequence of letters.

THEOREM 17C There is an effective procedure that, given a ﬁnite set ; τ of wffs, will decide whether or not |= τ .

PROOF. The truth-table procedure (Section 1.2) meets the requirement. In this theorem we speciﬁed that ; τ was ﬁnite, since one cannot be “given” in any direct and effective way all of an inﬁnite object.

COROLLARY 17D For a ﬁnite set , the set of tautological consequences of is decidable. In particular, the set of tautologies is decidable.

If is an inﬁnite set — even a decidable one — then in general its set of tautological consequences may not be decidable. (See Chapter 3.) But we can obtain a weaker result, which is in a sense half of decidability. Say that a set A of expressions is effectively enumerable iff there exists an effective procedure that lists, in some order, the members of A. If A is inﬁnite, then the procedure can never ﬁnish. But for any speciﬁed member of A, it must eventually (i.e., in a ﬁnite length of time) appear on the list. To give more of a feeling for this concept, we now state an equivalent way of formulating it.

THEOREM 17E A set A of expressions is effectively enumerable iff there exists an effective procedure that, given any expression ε, produces the answer “yes” iff ε ∈ A.

If ε ∈ / A, the procedure might produce the answer “no”; more likely it will go on forever without producing any answer, but it must not lie to us and produce the answer “yes.” Such a procedure is called a semidecision procedure — it is half of a decision procedure:

DEFINITION. A set A of expressions is semidecidable iff there exists an effective procedure that, given any expression ε, produces the answer “yes” iff ε ∈ A.

Thus Theorem 17E states that a set is effectively enumerable iff it is semidecidable. PROOF. If A is effectively enumerable, then given any ε we can examine the listing of A as our procedure churns it out. If and when ε appears, we say “yes.” (Thus if ε ∈ / A, no answer is ever given. It is this that keeps A from being decidable. When ε has failed to occur among the ﬁrst 1010 enumerated members of A, there is in general no way of knowing whether ε ∈ / A — in which

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A Mathematical Introduction to Logic case one should give up looking — or whether ε will occur in the very next step.) Conversely, suppose that we have the procedure described in the theorem. We want to create a listing of A. The idea is to enumerate all expressions, and to apply our given procedure to each. But we must budget our time sensibly. It is easy enough to enumerate effectively all expressions: ε1 , ε2 , ε3 , . . . . Then proceed according to the following scheme: 1. Spend one minute testing ε1 , for membership in A (using the given procedure). 2. Spend two minutes testing ε1 , then two minutes testing ε2 . 3. Spend three minutes testing ε1 , three minutes testing ε2 , and three minutes testing ε3 . And so forth. Of course whenever our procedure produces a “yes” answer, we put the accepted expression on the output list. Thus any member of A will eventually appear on the list. (It will appear inﬁnitely many times, unless we modify the above instructions to check for duplication.) Clearly any decidable set is also semidecidable. (We get a semidecision procedure even if the “no” bulb has burned out.) Hence any any decidable set is effectively enumerable.

THEOREM 17F A set of expressions is decidable iff both it and its complement (relative to the set of all expressions) are effectively enumerable.

PROOF. Exercise 8. This result is sometimes called “Kleene’s theorem.” Observe that if sets A and B are effectively enumerable, then so are A ∪ B and A ∩ B (Exercise 11). The class of decidable sets is also closed under union and intersection, and it is in addition closed under complementation. Now for a more substantive result:

THEOREM 17G If is a decidable set of wffs, then the set of tautological consequences of is effectively enumerable.

PROOF. Actually it is enough for to be effectively enumerated; consider an enumeration σ1 , σ2 , σ3 , . . . .

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Given any wff τ , we can test (by truth tables) successively whether or not ∅ |= τ, {σ1 } |= τ, {σ1 , σ2 } |= τ, {σ1 , σ2 , σ3 } |= τ, and so forth. If any of these conditions is met, then we answer “yes.” Otherwise, we keep trying. This does produce an afﬁrmative answer whenever |= τ , by the corollary to the compactness theorem. Later on, we will want to use effective procedures to compute functions. We will say that a function f is effectively computable (or simply computable) iff there exists an effective procedure that, given an input x, will eventually produce the correct output f (x).

Exercises 1. Assume that every ﬁnite subset of is satisﬁable. Show that the same is true of at least one of the sets ; α and ; ¬ α. (This is part of the proof of the compactness theorem.) Suggestion: If not, then 1 ; α and 2 ; ¬ α are unsatisﬁable for some ﬁnite 1 ⊆ and 2 ⊆ . Look at 1 ∪ 2 . 2. Let be a set of wffs such that (i) every ﬁnite subset of is satisﬁable, and (ii) for every wff α, either α ∈ or (¬ α) ∈ . Deﬁne the truth assignment v: T iff A ∈ , v(A) = F iff A ∈ / for each sentence symbol A. Show that for every wff ϕ, v(ϕ) = T iff ϕ ∈ . (This is part of the proof of the compactness theorem.) Suggestion: Use induction on ϕ. 3. Show that from the corollary to the compactness theorem we can prove the compactness theorem itself (far more easily than we can starting from scratch). 4. In 1977 it was proved that every planar map can be colored with four colors. Of course, the deﬁnition of “map” requires that there be only ﬁnitely many countries. But extending the concept, suppose we have an inﬁnite (but countable) planar map with countries C1 , C2 , C3 , . . . . Prove that this inﬁnite planar map can still be colored with four colors. (Suggestion: Partition the sentence symbols into four parts. One sentence symbol, for example, can be used to translate, “Country C7 is colored red.” Form a set 1 of wffs that say, for example, C7 is exactly one of the colors. Form another set

66

A Mathematical Introduction to Logic 2 of wffs that say, for each pair of adjacent countries, that they are not the same color. Apply compactness to 1 ∪ 2 .) 5. Where is a set of wffs, deﬁne a deduction from to be a ﬁnite sequence α0 , . . . , αn of wffs such that for each k ≤ n, either (a) αk is a tautology, (b) αk ∈ , or (c) for some i and j less than k, αi is (α j → αk ). (In case (c), one says that αk is obtained by modus ponens from αi and α j .) Give a deduction from the set {¬ S ∨ R, R → P, S} the last component of which is P. 6. Let α0 , . . . , αn be a deduction from a set of wffs, as in the preceding problem. Show that |= αk for each k ≤ n. Suggestion: Use strong induction on k, so that the inductive hypothesis is that |= αi for all i < k. 7. Show that whenever |= τ , then there exists a deduction from , the last component of which is τ . Remark: This result is called “completeness”; the concepts in Exercises 5–7 will reappear in Section 2.4. 8. Prove Theorem 17F. Remark: Two semidecision procedures make a whole. ∗

9. The concepts of decidability and effective enumerability can be applied not only to sets of expressions but also to sets of integers or to sets of pairs of expressions or integers. Show that a set A of expressions is effectively enumerable iff there is a decidable set B of pairs α, n (consisting of an expression α and an integer n) such that A = dom B. 10. Let be an effectively enumerable set of wffs. Assume that for each wff τ , either |= τ or |= ¬ τ . Show that the set of tautological consequences of is decidable. (a) Do this where “or” is interpreted in the exclusive sense: either |= τ or |= ¬ τ but not both. (b) Do this where “or” is interpreted in the inclusive sense: either |= τ or |= ¬ τ or both. 11. (a) Explain why the union of two effectively enumerable sets is again effectively enumerable. (b) Explain why the intersection of two effectively enumerable sets is again effectively enumerable. 12. For each of the following conditions, give an example of an unsatisﬁable set of formulas that meets the condition. (a) Each member of is — by itself — satisﬁable. (b) For any two members γ1 and γ2 of , the set {γ1 , γ2 } is satisﬁable. (c) For any three members γ1 , γ2 , and γ3 of , the set {γ1 , γ2 , γ3 } is satisﬁable.

Chapter

T W O

First-Order Logic SECTION 2.0 Preliminary Remarks The preceding chapter presented the ﬁrst of our mathematical models of deductive thought. It was a simple model, indeed too simple. It is easy to think of examples of intuitively correct deductions that cannot be adequately mirrored in the model of sentential logic. Suppose we begin with a collection of hypotheses (in English) and a possible conclusion. By translating everything to the language of sentential logic we obtain a set of hypotheses and a possible conclusion τ . Now if |= τ , then we feel that the original English-language deduction was valid. But if |= τ , then we are unsure. It may well be that the model of sentential logic was simply too crude to mirror the subtlety of the original deduction. This chapter presents a system of logic of much greater ability. In fact, when the “working mathematician” ﬁnds a proof, almost invariably what is meant is a proof that can be mirrored in the system of this chapter. First, we want to give an informal description of the features our ﬁrst-order languages might have (or at least might be able to simulate). We begin with a special case, the ﬁrst-order language for number theory. For this language there is a certain intended way of translating to and from English (Table VII).

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A Mathematical Introduction to Logic TABLE VII Formal expression

Intended translation

0

“Zero.” Here 0 is a constant symbol, intended to name the number 0. “The successor of t.” Here S is a one-place function symbol. t is to be an expression that names some number a. Then St names S(a), the successor of a. For example, S0 is intended to name the number 1. “v1 is less than v2 .” Here < is a two-place predicate symbol. At the end of Section 2.1 we will adopt conventions letting us abbreviate the expression in the more usual style: v1 < v2 . “For every natural number.” The symbol ∀ is the universal quantiﬁer symbol. More generally, with each translation of the language into English there will be associated a certain set A (the so-called universe); ∀ will then become “for every member of the universe A.” “For every natural number v1 , zero is less than v1 .” Or more euphoniously, “Every natural number is larger than 0.” This formal sentence is false in the intended translation, since zero is not larger than itself.

St

∀

∀ v1 < 0v1

One abbreviation is mentioned in Table VII. There will be more (Table VIII). Actually we will not be quite as generous as the tables might suggest. There are two economy measures that we can take to obtain simpliﬁcation without any essential loss of linguistic expressiveness: First, we choose as our sentential connective symbols just ¬ and →. We know from Section 1.5 that these form a complete set, so there is no compelling reason to use more. Secondly, we forego the luxury of an existential quantiﬁer, ∃ x. In its place we use ¬ ∀ x ¬. This is justiﬁed, since an English sentence like There is something rotten in the state of Denmark, is equivalent to It is not the case that for every x, x is not rotten in the state of Denmark. Thus the formula ∃ v1 ∀ v2 v1 = v2 becomes, in unabbreviated form, (¬ ∀ v1 (¬ ∀ v2 = v1 v2 )).

Chapter 2:

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69 TABLE VIII

Abbreviated expression

Intended translation

x=y

“x equals y.” In unabbreviated form this will become =x y. “There exists a natural number v such that.” Or more generally, “there exists a member of the universe such that.” “There is exactly one natural number.” Again this formal sentence is false in the intended translation. “Every natural number is greater than or equal to zero.”

∃v

∃ v1 ∀ v2 v1 = v 2

∀ v1 (0 < v1 ∨ 0 = v1 )

For an example in an ad hoc language, we might translate “Socrates is a man” as Hs, where H is a one-place predicate symbol intended to translate “is a man” and s is a constant symbol intended to name Socrates. Similarly, to translate “Socrates is mortal” we take Ms. Then “All men are mortal” is translated as ∀ v1 (H v1 → Mv1 ). The reader will possibly recognize the symbols ∀ and ∃ from previous mathematical contexts. Indeed, some mathematicians, when writing on the blackboard during their lectures, already use a nearly formalized language with only vestigial traces of English. That our ﬁrst-order languages resemble theirs is no accident. We want to be able to take one step back and study not, say, sets or groups, but the sentences of set theory or group theory. (The term metamathematics is sometimes used; the word itself suggests the procedure of stepping back and examining what the mathematician is doing.) The objects you, the logician, now study are the sentences that you, the set theoretician, previously used in the study of sets. This requires formalizing the language of set theory. And we want our formal languages to incorporate the features used in, for example, set theory.

SECTION 2.1 First-Order Languages We assume henceforth that we have been given inﬁnitely many distinct objects (which we call symbols), arranged as follows: A. Logical symbols 0. Parentheses: ( , ). 1. Sentential connective symbols: →, ¬.

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A Mathematical Introduction to Logic 2. Variables (one for each positive integer n): v1 , v2 , . . . . 3. Equality symbol (optional): =. B. Parameters 0. Quantiﬁer symbol: ∀. 1. Predicate symbols: For each positive integer n, some set (possibly empty) of symbols, called n-place predicate symbols. 2. Constant symbols: Some set (possibly empty) of symbols. 3. Function symbols: For each positive integer n, some set (possibly empty) of symbols, called n-place function symbols. In A.3 we allow for the possibility of the equality symbol’s being present, but we do not assume its presence. Some languages will have it and others will not. The equality symbol is a two-place predicate symbol but is distinguished from the other two-place predicate symbols by being a logical symbol rather than a parameter. (This status will affect its behavior under translations into English.) We do need to assume that some n-place predicate symbol is present for some n. In B.2, the constant symbols are also called 0-place function symbols. This will often allow a uniform treatment of the symbols in B.2 and B.3. As before, we assume that the symbols are distinct and that no symbol is a ﬁnite sequence of other symbols. In order to specify which language we have before us (as distinct from other ﬁrst-order languages), we must (i) say whether or not the equality symbol is present, and (ii) say what the parameters are. We now list some examples of what this language might be: 1. Pure predicate language Equality: No. n-place predicate symbols: An1 , An2 , . . . . Constant symbols: a1 , a2 , . . . . n-place function symbols (n > 0): None. 2. Language of set theory Equality: Yes (usually). Predicate parameters: One two-place predicate symbol ∈. Function symbols: None (or occasionally a constant symbol ∅). 3. Language of elementary number theory (as in Chapter 3) Equality: Yes. Predicate parameters: One two-place predicate symbol <. Constant symbols: The symbol 0.

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One-place function symbols: S (for successor). Two-place function symbols: + (for addition), · (for multiplication), and E (for exponentiation). In examples 2 and 3 there are certain intended translations of the parameters. We will presently give a number of examples of sentences that can be translated into these languages and a few examples of sentences that cannot be so translated. It is important to notice that our notion of language includes the language for set theory. For it is generally agreed that, by and large, mathematics can be embedded into set theory. By this is meant that (a) Statements in mathematics (such as the fundamental theorem of calculus) can be expressed in the language of set theory; and (b) The theorems of mathematics follow logically from the axioms of set theory. Our model of ﬁrst-order logic is fully adequate to mirror this procedure. EXAMPLES in the language of set theory. Here it is intended that ∀ should mean “for all sets” and ∈ should mean “is a member of.” 1. “There is no set of which every set is a member.” We will translate this into the language of set theory using several steps. The intermediate sentences are neither in English nor in the formal language but are in a mixed language. ¬[There is a set of which every set is a member] ¬ ∃ v1 [Every set is a member of v1 ] ¬ ∃ v1 ∀ v2 v2 ∈ v1 Although it is tempting to stop here, we must now replace v2 ∈ v1 by ∈v2 v1 , since predicate symbols will always go at the left in such contexts. Furthermore, ∃ v1 must be replaced by ¬ ∀ v1 ¬, as mentioned earlier. And we must use the correct number of parentheses. The ﬁnished product is (¬ (¬ ∀ v1 (¬ ∀ v2 ∈v2 v1 ))). 2. Pair-set axiom: “For any two sets, there is a set whose members are exactly the two given sets.” Again we carry out the translation in stages. ∀ v1 ∀ v2 [There is a set whose members are exactly v1 and v2 ] ∀ v1 ∀ v2 ∃ v3 [The members of v3 are exactly v1 and v2 ] ∀ v1 ∀ v2 ∃ v3 ∀ v4 (v4 ∈ v3 ↔ v4 = v1 ∨ v4 = v2 ) Now we replace ∃ v3 by ¬ ∀ v3 ¬, v4 ∈ v3 by ∈v4 v3 , and v4 = vi by =v4 vi . In addition, we must eliminate ↔ and ∧ in favor of our chosen connectives → and ¬. Thus α ∨ β becomes ¬ α → β; α ↔ β becomes ¬ ((α → β) → ¬ (β → α)).

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A Mathematical Introduction to Logic The ﬁnished product is ∀ v1 ∀ v2 (¬ ∀ v3 (¬ ∀ v4 (¬ ((∈v4 v3 → (( ¬ =v4 v1 )→ = v4 v2 )) → (¬ (((¬ =v4 v1 ) → =v4 v2 ) → ∈v4 v3 )))))). The ﬁnished product is not nearly as pleasant to read as the version that preceded it. As we have no interest in deliberately making life unpleasant for ourselves, we will eventually adopt conventions allowing us to avoid seeing the ﬁnished product at all. But for the moment it should be regarded as an interesting, even if unattractive, novelty. EXAMPLES in the language of elementary number theory. Here it is intended that ∀ should mean “for all natural numbers” and that <, 0, S, +, ·, and E should have the obvious meanings. 1. As a name for the natural number 2 we have the term SS0, since 2 is the successor of the successor of zero. Similarly, for 4 we have the term SSSS0. For the phrase “2 + 2” it is tempting to use SS0 + SS0. But we will adopt the policy of always putting the function symbol at the left (i.e., we will use Polish notation for function symbols). Thus corresponding to the English phrase “2 + 2” we have the term + SS0 SS0. The English sentence “Two plus two is four” is translated as = + SS0 SS0 SSSS0. (The spaces are inserted to help the reader, but they do not constitute an ofﬁcial feature of the language.) 2. “Any nonzero natural number is the successor of some number.” We will perform the translation in three steps: ∀ v1 [ If v1 is nonzero, then v1 is the successor of some number.] 0 → ∃ v2 v1 = Sv2 ). ∀ v1 (v1 = ∀ v1 ((¬ =v1 0) → (¬ ∀ v2 (¬ =v1 Sv2 ))). 3. “Any nonempty set of natural numbers has a least element.” This cannot be translated into our language, because we cannot express “any . . . set.” This requires either something like the (ﬁrstorder) language for set theory or a second-order language for number theory. We could, however, translate, “The set of primes has a least element.” (The ﬁrst step is to convert this sentence into, “There is a smallest prime.” We leave the other steps to the reader; hints can be found in the next section.) EXAMPLES

in ad hoc languages

1. “All apples are bad.” ∀ v1 (Av1 → Bv1 ).

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2. “Some apples are bad.” Intermediate step: ∃ v1 (Av1 ∧ Bv1 ). Finished product: (¬ ∀ v1 (¬ (¬ (Av1 → (¬ Bv1 ))))). These two examples illustrate patterns that arise continually. An English sentence which asserts that everything in a certain category has some property is translated ∀ v(

→

).

A sentence which asserts that there is some object or objects in the category and having the property is translated ∃ v(

∧

).

The reader should be cautioned against confusing the two patterns. For example, ∀ v1 (Av1 ∧ Bv1 ) translates “Everything is an apple and is bad,” which is a much stronger assertion than the sentence in the ﬁrst example. Similarly, ∃ v1 (Av1 → Bv1 ) translates “There is something which is bad, if it is an apple.” This is a much weaker assertion than the sentence in the second example. It is true (vacuously) even if all apples are good, provided only that the world has something which is not an apple. 3. “Bobby’s father can beat up the father of any other kid on the block.” Establish a language in which ∀ is intended to mean “for all people,” K x is to mean “x is a kid on the block,” b is to mean “Bobby,” Bx y is to mean “x can beat up y,” and f x is to mean “the father of x.” Then a translation is ∀ v1 (K v1 → ((¬ =v1 b) → B f b f v1 )). 4. In calculus, we learn the meaning of “the function f converges to L as x approaches a”: ∀ ε(ε > 0 → ∃ δ(δ > 0 ∧ ∀ x(|x − a| < δ → | f x − L| < ε))). This is, apart from notational matters, a formula of the sort we are interested in, using a predicate symbol for ordering, function symbols for f , subtraction, and absolute values, and constant symbols for 0, a, and L.

Formulas An expression is any ﬁnite sequence of symbols. Of course most expressions are nonsensical, but there are certain interesting expressions: the terms and the wffs.

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A Mathematical Introduction to Logic The terms are the nouns and pronouns of our language; they are the expressions that can be interpreted as naming an object. The atomic formulas will be those wffs having neither connective nor quantiﬁer symbols.

The terms are deﬁned to be those expressions that can be built up from the constant symbols and the variables by preﬁxing the function symbols. To restate this in the terminology of Chapter 1, we deﬁne for each n-place function symbol f , an n-place term-building operation F f on expressions: F f (ε1 , . . . , εn ) = f ε1 · · · εn . DEFINITION. The set of terms is the set of expressions that can be built up from the constant symbols and variables by applying (zero or more times) the F f operations. If there are no function symbols (apart from the constant symbols), then the terms are just the constant symbols and the variables. In this case we do not need an inductive deﬁnition. Notice that we use Polish notation for terms by placing the function symbol at the left. The terms do not contain parentheses or commas. We will later prove a unique readability result, showing that given a term, we can unambiguously decompose it. The terms are the expressions that are translated as names of objects (noun phrases), in contrast to the wffs which are translated as assertions about objects. Some examples of terms in the language of number theory are +v2 S0, SSSS0, +Ev1 SS0 Ev2 SS0. The atomic formulas will play a role roughly analogous to that played by the sentence symbols in sentential logic. An atomic formula is an expression of the form Pt1 · · · tn , where P is an n-place predicate symbol and t1 , . . . , tn are terms. For example, =v1 v2 is an atomic formula, since = is a two-place predicate symbol and each variable is a term. In the language of set theory we have the atomic formula ∈v5 v3 .

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Notice that the atomic formulas are not deﬁned inductively. Instead we have simply said explicitly just which expressions are atomic formulas. The well-formed formulas are those expressions that can be built up from the atomic formulas by use (zero or more times) of the connective symbols and the quantiﬁer symbol. We can restate this in the terminology of Chapter 1 by ﬁrst deﬁning some formula-building operations on expressions: E¬ (γ ) = (¬ γ ), E→ (γ , δ) = (γ → δ), Qi (γ ) = ∀ vi γ . DEFINITION. The set of well-formed formulas (wffs, or just formulas) is the set of expressions that can be built up from the atomic formulas by applying (zero or more times) the operations E¬ , E → , and Qi (i = 1, 2, . . .). For example, on the one hand, ¬ v3 is not a wff. (Why?) On the other hand, ∀ v1 ((¬ ∀ v3 ( ¬ ∈v3 v1 )) → (¬ ∀ v2 (∈v2 v1 → (¬ ∀ v4 (∈v4 v2 → ( ¬ ∈v4 v1 )))))) is a wff, as is demonstrated by the following tree:

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A Mathematical Introduction to Logic But it requires some study to discover that this wff is the axiom of regularity for set theory.

Free Variables Two examples of wffs are ∀ v2 ∈v2 v1 and (¬ ∀ v1 (¬ ∀ v2 ∈v2 v1 )). But there is an important difference between the two examples. The second might be translated back into English as There is a set such that every set is a member of it. The ﬁrst example, however, can be translated only as an incomplete sentence, such as Every set is a member of

1.

We are unable to complete the sentence without knowing what to do with v1 . In cases of this sort, we will say that v1 occurs free in the wff ∀ v2 ∈v2 v1 . In contrast, no variable occurs free in (¬ ∀ v1 (¬ ∀ v2 ∈v2 v1 )). But of course we need a precise deﬁnition which does not refer to possible translations to English but refers only to the symbols themselves. Consider any variable x. We deﬁne, for each wff α, what it means for x to occur free in α. This we do by recursion: 1. For atomic α, x occurs free in α iff x occurs in (i.e., is a symbol of) α. 2. x occurs free in (¬ α) iff x occurs free in α. 3. x occurs free in (α → β) iff x occurs free in α or in β. 4. x occurs free in ∀ vi α iff x occurs free in α and x =

vi . This deﬁnition makes tacit use of the recursion theorem. We can restate the situation in terms of functions. We begin with the function h deﬁned on atomic formulas: h(α) = the set of all variables, if any, in the atomic formula α. And we want to extend h to a function h deﬁned on all wffs in such a way that h(E¬ (α)) = h(α), h(E→ (α, β)) = h(α) ∪ h(β), h(Qi (α)) = h(α) after removing vi , if present. Then we say that x occurs free in α (or that x is a free variable of α) iff x ∈ h(α). The existence of a unique such h (and hence the meaningfulness of our deﬁnition) follows from the recursion theorem of Section 1.4 and from the fact (proved in Section 2.3) that each wff has a unique decomposition.

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If no variable occurs free in the wff α (i.e., if h(α) = ∅), then α is a sentence. (The sentences are intuitively the wffs translatable into English without blanks, once we are told how to interpret the parameters.) For example, ∀ v2 (Av2 → Bv2 ) and ∀ v3 (Pv3 → ∀ v3 Qv3 ) are sentences, but v1 occurs free in ( ∀ v1 Av1 →Bv1 ). The sentences are usually the most interesting wffs. The others lead a second-class existence; they are used primarily as building blocks for sentences. In translating a sentence from English, the choice of particular variables is unimportant. We earlier translated “All apples are bad” as ∀ v1 (Av1 → Bv1 ). We could equally well have used ∀ v27 (Av27 → Bv27 ). The variable is, in effect, used as a pronoun, just as in English we might say, “For any object whatsoever, if it is an apple, then it is bad.” (We have incorporated into our language an adequate supply of pronouns: it1 , it2 , . . . .) Since the choice of particular variables is unimportant, we will often not even specify the choice. Instead we will write, for example, ∀ x(Ax → Bx), where it is understood that x is some variable. (The unimportance of the choice of variable will eventually become a theorem.) Similar usages of variables occur elsewhere in mathematics. In 7

ai j

i=1

i is a “dummy” variable but j occurs free.

On Notation We can specify a wff (or indeed any expression) by writing a line that displays explicitly every symbol. For example, ∀ v1 ((¬ =v1 0) → (¬ ∀ v2 (¬ =v1 Sv2 ))). But this way of writing things, while splendidly complete, may not be readily comprehensible. The incomprehensibility can be blamed (in part) on the simpliﬁcations we wanted in the language (such as the lack of an existential quantiﬁer symbol). We naturally want to have our cake and eat it too, so we now will agree on methods of specifying wffs in more indirect but more readable ways. These conventions will let us write a line such as 0 → ∃ v2 v1 = Sv2 ) ∀ v1 (v1 = to name the same wff as is named by the other line above. Note well that we are not changing our deﬁnition of what a wff is. We are simply conspiring to ﬁx certain ways of naming wffs. In the (rare) cases where the exact sequence of symbols becomes important,

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A Mathematical Introduction to Logic we may have to drop these new conventions and revert to primitive notation. We adopt then the following abbreviations and conventions. Here α and β are formulas, x is a variable, and u and t are terms. (α ∨ β) abbreviates ((¬ α) → β). (α ∧ β) abbreviates ( ¬ (α → ( ¬ β))). (α ↔ β) abbreviates ((α → β) ∧ (β → α)); i.e., ( ¬ ((α → β) → ( ¬ (β → α)))). ∃ x α abbreviates ( ¬ ∀ x( ¬ α)). u = t abbreviates =ut. A similar abbreviation applies to some other two-place predicate and function symbols. For example, 2 < 3 abbreviates <2 3 and 2 + 2 abbreviates +2 2. u= t abbreviates (¬ =ut); similarly u < t abbreviates ( ¬

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Predicate symbols: Uppercase italic letters. Also ∈, <. Variables: vi , u, v, x, y, z. Function symbols: f, g, h. Also S, +, etc. Constant symbols: a, b, . . . . Also 0. Terms: u, t. Formulas: Lowercase Greek letters. Sentences: σ, τ . Sets of formulas: Uppercase Greek letters, plus certain italic letters that pretend to be Greek, viz., A (alpha) and T (tau). Structures (see the next section): Uppercase German (Fraktur) letters.

Exercises 1. Assume that we have a language with the following parameters: ∀, intended to mean “for all things”; N , intended to mean “is a number”; I , intended to mean “is interesting”; <, intended to mean “is less than”; and 0, a constant symbol intended to denote zero. Translate into this language the English sentences listed below. If the English sentence is ambiguous, you will need more than one translation. (a) Zero is less than any number. (b) If any number is interesting, then zero is interesting. (c) No number is less than zero. (d) Any uninteresting number with the property that all smaller numbers are interesting certainly is interesting. (e) There is no number such that all numbers are less than it. (f) There is no number such that no number is less than it. 2. With the same language as in the preceding exercise, translate back into good English the wff ∀ x(N x → I x → ¬ ∀ y(N y → I y → ¬ x < y)). In 3–8, translate each English sentence into the ﬁrst-order language speciﬁed. (You may want to carry out the translation in several steps, as in some of the examples.) Make full use of the notational conventions and abbreviations to make the end result as readable as possible. 3. Neither a nor b is a member of every set. (∀, for all sets; ∈, is a member of; a, a; b, b.) 4. If horses are animals, then heads of horses are heads of animals. (∀, for all things; E, is a horse; A, is an animal; hx, the head of x.) 5. (a) You can fool some of the people all of the time. (b) You can fool all of the people some of the time. (c) You can’t fool all of the people all of the time. (∀, for all things; P, is a person; T , is a

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A Mathematical Introduction to Logic time; F x y, you can fool x at y. One or more of the above may be ambiguous, in which case you will need more than one translation.) 6. (a) Adams can’t do every job right. (b) Adams can’t do any job right. (∀, for all things; J , is a job; a, Adams; Dx y, x can do y right.) 7. (a) Nobody likes everybody. (b) No Democrat likes every Republican. (∀, for all people; L x y, x likes y; D, is a Democrat; R, is a Republican.) 8. (a) Every farmer who owns a donkey needs hay. (b) Every farmer who owns a donkey beats it. (∀, for all things; F, is a farmer; D, is a donkey; O x y, x owns y; H , needs hay; Bx y, x beats y.) 9. Give a precise deﬁnition of what it means for the variable x to occur free as the ith symbol in the wff α. (If x is the ith symbol of α but does not occur free there, then it is said to occur bound there.) 10. Rewrite each of the following wffs in a way which explicitly lists each symbol in the actual order: (a) ∃ v1 Pv1 ∧ Pv1 . (b) ∀ v1 Av1 ∧ Bv1 → ∃ v2 ¬ Cv2 ∨ Dv2 . In each case, say which variables occur free in the wff.

SECTION 2.2 Truth and Models In sentential logic we had truth assignments to tell us which sentence symbols were to be interpreted as being true and which as false. In ﬁrstorder logic the analogous role is played by structures, which can be thought of as providing the dictionary for translations from the formal language into English. (Structures are sometimes called interpretations, but we prefer to reserve that word for another concept, to be encountered in Section 2.7.) A structure for a ﬁrst-order language will tell us 1. What collection of things the universal quantiﬁer symbol (∀) refers to, and 2. What the other parameters (the predicate and function symbols) denote. Formally, a structure A for our given ﬁrst-order language is a function whose domain is the set of parameters and such that1

1

The symbol “A” is the letter A in the German (Fraktur) alphabet. The next two letters are B and C.

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1. A assigns to the quantiﬁer symbol ∀ a nonempty set |A| called the universe (or domain) of A. 2. A assigns to each n-place predicate symbol P an n-ary relation P A ⊆ |A|n ; i.e., P A is a set of n-tuples of members of the universe. 3. A assigns to each constant symbol c a member cA of the universe |A|. 4. A assigns to each n-place function symbol f an n-ary operation f A on |A|; i.e., f A : |A|n → |A|. The idea is that A assigns meaning to the parameters. ∀ is to mean “for everything in |A|.” The symbol c is to name the point cA . The atomic formula Pt1 · · · tn is to mean that the n-tuple of points named by t1 , . . . , tn is in the relation P A . (We will shortly restate these conditions more carefully.) Note that we require the universe |A| to be nonempty. Notice also that f A must have all of |A|n in for its domain; we have made no provision for partially deﬁned functions. EXAMPLE. Consider the language for set theory, whose only parameter (other than ∀) is ∈. Take the structure A with |A| = the set of natural numbers, ∈A = the set of pairs m, n such that m < n. (Thus we translate ∈ as “is less than.”) In the presence of a structure we can translate sentences from the formal language into English and attempt to say whether these translations are true or false. The sentence of this ﬁrst-order language ∃x ∀y¬y ∈ x (or more formally, (¬ ∀ v1 (¬ ∀ v2 ( ¬ ∈v2 v1 )))), which under another translation asserts the existence of an empty set, is now translated under A into There is a natural number such that no natural number is smaller, which is true. Because of this we will say that ∃ x ∀ y ¬ y ∈ x is true in A, or that A is a model of the sentence. On the other hand, A is not a model of the pair-set axiom, ∀ x ∀ y ∃ z ∀ t(t ∈ z ↔ t = x ∨ t = y), as the translation of this sentence under A is false. For there is no natural number m such that for every n, n

iff n = 1.

(The reader familiar with axiomatic set theory can check that A

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A Mathematical Introduction to Logic is a model of the extensionality axiom, the union axiom, and the axiom of regularity.) EXAMPLE. Again assume the language has only the parameters ∀ and a two-place predicate symbol E. But this time, consider the ﬁnite structure B with universe |B| consisting of a set of four distinct objects {a, b, c, d}. Suppose the binary relation E B is the following set of pairs: E B = { a, b , b, a , b, c , c, c }. Then we can picture B as the directed graph whose vertex-set is the universe {a, b, c, d}: a

b

c

d

Here we interpret E x y as saying there is an edge from vertex x to vertex y in the graph. (If the binary relation E B had been symmetric, then we could have pictured the structure as an undirected graph.) Consider now the sentence ∃ x ∀ y ¬ y E x. Under the structure B, we can translate this as follows: There is a vertex such that for any vertex, no edge points from the latter to the former. (The English version is harder to read than the symbolic one!) This sentence is true in B, because no edge points to the vertex d. In the preceding examples it was intuitively pretty clear that certain sentences of the formal language were true in the structure and some were false. But we want a precise mathematical deﬁnition of “σ is true in A.” This should be stated in mathematical terms, without employing translations into English or a supposed criterion for asserting that some English sentences are true while the others are false. (If you think you have such a criterion, try it on the sentence “This sentence is false.”) In other words, we want to take our informal concept of “σ is true in A” and make it part of mathematics. In order to deﬁne “σ is true in A,” |=A σ, for sentences σ and structures A, we will ﬁnd it desirable ﬁrst to deﬁne a more general concept involving wffs. Let ϕ be a wff of our language, A a structure for the language,

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s : V → |A| a function from the set V of all variables into the universe |A| of A. Then we will deﬁne what it means for A to satisfy ϕ with s, |=A ϕ[s]. The informal version is |=A ϕ[s] if and only if the translation of ϕ determined by A, where the variable x is translated as s(x) wherever it occurs free, is true. The formal deﬁnition of satisfaction proceeds as follows: I. Terms. We deﬁne the extension s : T → |A|, a function from the set T of all terms into the universe of A. The idea is that s(t) should be the member of the universe |A| that is named by the term t. s is deﬁned by recursion as follows: 1. For each variable x, s(x) = s(x). 2. For each constant symbol c, s(c) = cA . 3. If t1 , . . . , tn are terms and f is an n-place function symbol, then s( f t1 · · · tn ) = f A (s(t1 ), . . . , s(tn )). A commutative diagram, for n = 1, is

The existence of a unique such extension s of s follows from the recursion theorem (Section 1.4), by using the fact that the terms have unique decompositions (Section 2.3). Notice that s depends both on s and on A. (In fact a reasonable alternative notation for s(t) would be t A [s], which explicitly displays the dependence on A.) II. Atomic formulas. The atomic formulas were deﬁned explicitly, not inductively. The deﬁnition of satisfaction of atomic formulas is therefore also explicit, and not recursive. 1. |=A =t1 t2 [s] iff s(t1 ) = s(t2 ). (Thus = means =. Note that = is a logical symbol, not a parameter open to interpretation.) 2. For an n-place predicate parameter P, |=A Pt1 · · · tn [s] iff s(t1 ), . . . , s(tn ) ∈ P A .

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A Mathematical Introduction to Logic III. Other wffs. The wffs we deﬁned inductively, and consequently here satisfaction is deﬁned recursively. 1. For atomic formulas, the deﬁnition is above. 2. |=A ¬ ϕ[s] iff |=A ϕ[s]. 3. |=A (ϕ → ψ)[s] iff either |=A ϕ[s] or |=A ψ[s] or both. (In other words, if A satisﬁes ϕ with s then A satisﬁes ψ with s.) 4. |=A ∀x ϕ[s] iff for every d ∈ |A|, we have |=A ϕ[s(x | d)]. Here s(x | d) is the function which is exactly like s except for one thing: At the variable x it assumes the value d. This can be expressed by the equation s(y) if y =

x, s(x | d)(y) = d if y = x. (Thus ∀ means “for all things in |A|.”) At this point the reader might want to reconsider the informal version of |=A ϕ[s] on page 83 and observe how it has been formalized. We should remark that the deﬁnition of satisfaction is another application of the recursion theorem together with the fact that the wffs have unique decompositions. The deﬁnition can be restated in terms of functions to make it clearer how the recursion theorem of Section 1.4 applies: (i) Consider one ﬁxed A. (ii) Deﬁne a function h (extending a function h deﬁned on atomic formulas) such that for any wff ϕ, h(ϕ) is a set of functions from V into |A|. (iii) Deﬁne |=A ϕ[s]

iff

s ∈ h(ϕ).

We leave to the reader the exercise of writing down the explicit deﬁnition of h and the clauses that uniquely determine its extension h. (See Exercise 7.) An elegant alternative is to have h(ϕ) be a set of functions on the set of those variables that occur free in ϕ. EXAMPLE. Assume that our language has the parameters ∀, P (a twoplace predicate symbol), f (a one-place function symbol), and c (a constant symbol). Let A be the structure for his language deﬁned as follows: |A| = N, the set of all natural numbers, P A = the set of pairs m, n such that m ≤ n, f A = the successor function S; f A (n) = n + 1, cA = 0. We can summarize this in one line, by suppressing the fact that A

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is really a function and merely listing its components: A = (N; ≤, S, 0). This notation is unambiguous only when the context makes clear just which components go with which parameters. Let s : V → N be the function for which s(vi ) = i − 1; i.e., s(v1 ) = 0, s(v2 ) = 1, and so forth. 1. s( f f v3 ) = S(S(2)) = 4 and s( f v1 ) = S(0) = 1. 2. s(c) = 0 and s( f f c) = 2; no use is made of s. 3. |=A Pc f v1 [s]. This is informally obvious, since when we translate back into English we get the true sentence “0 ≤ 1.” More formally, the reason is that s(c), s( f v1 ) = 0, 1 ∈ P A . 4. |=A ∀ v1 Pcv1 . The translation into English is “0 is less than or equal to any natural number.” Formally we must verify that for any n in N, |=A Pcv1 [s(v1 | n)], which reduces to 0, n ∈ P A , i.e., 0 ≤ n. 5. |=A ∀ v1 Pv2 v1 [s] because there is a natural number m such that

|=A Pv2 v1 [s(v1 | m)]; i.e., / P A. s(v2 ), m ∈ In fact, since s(v2 ) = 1, we must take m to be 0. The reader is to be cautioned against confusing, for example, the function symbol f with the function f A . EXAMPLE.

Previously we considered the structure B with

|B| = {a, b, c, d} and

E B = { a, b , b, a , b, c , c, c }

for the language with parameters ∀ and E: a

b

c

d

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A Mathematical Introduction to Logic Then |=B ∀ v2 ¬ Ev2 v1 [s] iff s(v1 ) = d. That is, there is no edge pointing to the vertex d, but d is the only such vertex. Taking the negation of the formula, we have |=B ∃ v2 Ev2 v1 [s] iff s(v1 ) ∈ {a, b, c}. At this point we pause to verify that when we want to know whether or not a structure A satisﬁes a wff ϕ with s, we do not really need all of the (inﬁnite amount of) information s gives us. All that matter are the values of the function s at the (ﬁnitely many) variables which occur free in ϕ. In particular, if ϕ is a sentence, then s does not matter at all. THEOREM 22A Assume that s1 and s2 are functions from V into |A| which agree at all variables (if any) that occur free in the wff ϕ. Then |=A ϕ[s1 ]

iff

|=A ϕ[s2 ].

PROOF. Because satisfaction was deﬁned recursively, this proof uses induction. We consider the ﬁxed structure A and show by induction that every wff ϕ has the property that whenever two functions s1 , s2 agree on the variables free in ϕ then A satisﬁes ϕ with s1 iff it does so with s2 . Case 1: ϕ = Pt1 · · · tn is atomic. Then any variable in ϕ occurs free. Thus s1 and s2 agree at all the variables in each ti . It follows that s 1 (ti ) = s 2 (ti ) for each i; a detailed proof would use induction on ti . Consequently, A satisﬁes Pt1 · · · tn with s1 iff it does so with s2 . Cases 2 and 3: ϕ has the form ¬ α or α → β. These cases are immediate from the inductive hypothesis. Case 4: ϕ = ∀ x ψ. Then the variables free in ϕ are those free in ψ with the exception of x. Thus for any d in |A|, s1 (x | d) and s2 (x | d) agree at all variables free in ψ. By inductive hypothesis, then, A satisﬁes ψ with s1 (x | d) iff it does so with s2 (x | d). From this and the deﬁnition of satisfaction we see that A satisﬁes ∀ x ψ with s1 iff it does so with s2 . In effect, the above proof amounts to looking through the deﬁnition of satisfaction and seeing what information given by s was actually used. There is an analogous fact regarding structures: If A and B agree at all the parameters that occur in ϕ, then |=A ϕ[s] iff |=B ϕ[s]. This theorem justiﬁes the following notation: Suppose that ϕ is a formula such that all variables occurring free in ϕ are included among v1 , . . . , vk . Then for elements a1 , . . . , ak of |A|, |=A ϕ[[a1 , . . . , ak ]]

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means that A satisﬁes ϕ with some (and hence with any) function s : V → |A| for which s(vi ) = ai , 1 ≤ i ≤ k. To return to a recent example where A = (N; ≤, S, 0), we have |=A ∀ v2 Pv1 v2 [[0]] but

|=A ∀ v2 Pv1 v2 [[5]]. COROLLARY 22B

For a sentence σ , either

(a) A satisﬁes σ with every function s from V into |A|, or (b) A does not satisfy σ with any such function. If alternative (a) holds, then we say that σ is true in A (written |=A σ ) or that A is a model of σ . And if alternative (b) holds, then of course σ is false in A. (They cannot both hold since |A| is nonempty). A is a model of a set of sentences iff it is a model of every member of . EXAMPLE. If R is the real ﬁeld, (R; 0, 1, +, ×), and Q is the rational ﬁeld, (Q; 0, 1, +, ×), is√there a sentence true in one and false in the other? Yes; because 2 is irrational, the sentence ∃ x(x · x = 1 + 1) is false in the rational ﬁeld, but true in the real ﬁeld. EXAMPLE. Suppose our given language has only the parameters ∀ and P, where P is a two-place predicate symbol. Then a structure A is determined by the universe |A| and the binary relation P A . With some minor illegality we again write A = (|A|; P A ). Now consider the problem of characterizing the class of all models of the following sentences: 1. ∀ x ∀ y x = y. A structure (A; R) is a model of this iff A contains exactly one element. R can either be empty or can be the singleton A × A. 2. ∀ x ∀ y P x y. A structure (A; R) is a model of this iff R = A × A. A can be any nonempty set. 3. ∀ x ∀ y ¬ P x y. A structure (A; R) is a model of this iff R = ∅. 4. ∀ x ∃ y P x y. The condition for (A; R) to be a model of this is that the domain of R is A. The notational conventions adopted earlier were done in a rational way: 1. |=A (α ∧ β)[s] iff |=A α[s] and |=A β[s]; similarly for ∨ and ↔. 2. |=A ∃ x α[s] iff there is some d ∈ |A| with the property that |=A α[s(x | d)].

88

A Mathematical Introduction to Logic The proof for the second of these is as follows: |=A ∃ x α[s]

iff |=A ¬ ∀ x ¬ α[s], iff |=A ∀ x ¬ α[s], iff it is not the case that for all d in |A|, |=A ¬ α[s(x | d)], iff it is not the case that for all d in |A|,

|=A α[s(x | d)], iff for some d in |A|, |=A α[s(x | d)].

Logical Implication We now have the necessary tools to formulate the important concept of logical implication for our language. DEFINITION. Let be a set of wffs, ϕ a wff. Then logically implies ϕ, written |= ϕ, iff for every structure A for the language and every function s : V → |A| such that A satisﬁes every member of with s, A also satisﬁes ϕ with s. We use the same symbol, “|=,” that was used in Chapter 1 for tautological implication. But henceforth it will be used solely for logical implication. As before we will write “γ |= ϕ” in place of “{γ } |= ϕ.” Say that ϕ and ψ are logically equivalent (ϕ |==| ψ) iff ϕ |= ψ and ψ |= ϕ. The ﬁrst-order analogue of the concept of a tautology is the concept of a valid formula: A wff ϕ is valid iff ∅ |= ϕ (written simply “|= ϕ”). Thus ϕ is valid iff for every A and every s : V → |A|, A satisﬁes ϕ with s. For sentences, the concept of logical implication can be stated more concisely, by applying Theorem 22A: COROLLARY 22C For a set ; τ of sentences, |= τ iff every model of is also a model of τ . A sentence τ is valid iff it is true in every structure. EXAMPLES of logical implication. Readers are invited to convince themselves of each of the following: 1. ∀ v1 Qv1 |= Qv2 . 2. Qv1 |= ∀ v1 Qv1 . Here it sufﬁces to ﬁnd just one structure A and just one function s : V → |A| such that on the one hand, |=A Qv1 [s] but, on the other hand, A is not a model of ∀ v1 Qv1 . |A| will need to have at least two members. 3. |= ¬ ¬ σ → σ . If A is a model of ¬ ¬ σ , then |=A ¬ σ whence |=A σ . But one might raise the objection: Aren’t we

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using here the law of double negation, the law we are claiming to prove? The answer is a deﬁnite yes and no. We are proving the law of double negation for the formal language we are talking about (sometimes called the object language). In doing so, we can of course use any correct reasoning (out in the meta-language, English), exactly as we might do in reasoning about vector spaces or graphs. In particular, the reasoning might involve principles that when formally modeled would involve ¬ ¬ σ and σ . There is no circularity. But the meta-language sentences we use are — unsurprisingly — related to the object language formulas we talk about. In this connection, see the book’s only picture (at the end of Section 2.4). 4. ∀ v1 Qv1 |= ∃ v2 Qv2 . Recall that the universe of any structure is nonempty. 5. ∃ x ∀ y P x y |= ∀ y ∃ x P x y. This example will come up again in Section 2.4. 6. ∀ y ∃ x P x y |= ∃ x ∀ y P x y. 7. |= ∃ x(Qx → ∀ x Qx). This is a strange — but valid — sentence. The deﬁnition of logical implication is very much like the deﬁnition of tautological implication in Chapter 1. But there is an important difference of complexity. Suppose in sentential logic you want to know whether or not a wff α is a tautology. The deﬁnition requires that you consider ﬁnitely many truth assignments, each of which is a ﬁnite function. For each such truth assignment v, you must calculate v(α), which can be effectively done in a ﬁnite length of time. (Consequently, the set of tautologies is decidable, as was observed before.) In contrast to the ﬁnitary procedure for tautologies, suppose that you want to know whether or not a wff ϕ (of our ﬁrst-order language) is valid. The deﬁnition requires that you consider every structure A. (In particular this requires using every nonempty set, of which there are a great many.) For each of these structures, you then must consider each function s from the set V of variables into |A|. And for each given A and s, you must determine whether or not A satisﬁes ϕ with s. When |A| is inﬁnite, this is a complicated notion in itself. In view of these complications, it is not surprising that the set of valid formulas fails to be decidable (cf. Section 3.5). What is surprising is that the concept of validity turns out to be equivalent to another concept (deducibility) whose deﬁnition is much closer to being ﬁnitary. (See Section 2.4.) Using that equivalence we will be able to show (under some reasonable assumptions) that the set of validities (i.e., the set of valid wffs) is effectively enumerable. The effective enumeration procedure yields a more concrete characterization of the set of valid formulas.

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Deﬁnability in a Structure Suppose we want to study the real ﬁeld, (R; 0, 1, +, ·), consisting of the set R of real numbers, together with the distinguished elements 0 and 1 and the two operations of addition and multiplication. We can consider the real ﬁeld as a structure R = (R; 0, 1, +, ·) where the language (with equality) has constant symbols 0 and 1 and two-place function symbols + and ·. Although we have not included an ordering symbol < in the language, we still have a way to say “x ≥ 0.” Because in this structure, the nonnegative elements are exactly the elements with square roots. That is, the formula ∃ v2 x = v2 · v2 is satisﬁed in the structure R whenever x is assigned a nonnegative number, and only then: |=R ∃ v2 v1 = v2 · v2 [[a]] ⇐⇒ a ≥ 0. Because of this fact, we will say that the interval [0, ∞) is deﬁnable in R, and that the formula ∃ v2 v1 = v2 · v2 deﬁnes it. Moreover, the ordering relation on the reals, i.e., the binary relation { a, b ∈ R × R | a ≤ b}, is deﬁned in the structure R by the formula expressing “v1 ≤ v2 ”: ∃v3 v2 = v1 + v3 · v3 . For a smaller example, take the directed graph A = ({a, b, c}; { a, b , a, c }) where the language has parameters ∀ and E: b

a

c

Then in A, the set {b, c} (the range of the relation E A ) is deﬁned by the formula ∃ v2 Ev2 v1 . In contrast, the set {b} is not deﬁnable in A. This is because there is no deﬁnable property in this structure that would separate b and c; the proof of this fact will utilize the homomorphism theorem, to be proved later in this section. We now want to set forth precisely this concept of deﬁnability of a subset of the universe or of a relation on the universe. Consider a structure A and a formula ϕ whose free variables are among v1 , . . . , vk . Then we can construct the k-ary relation on |A| { a1 , . . . , ak | |=A ϕ[[a1 , . . . , ak ]]}. Call this the k-ary relation ϕ deﬁnes in A. In general, a k-ary relation

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on |A| is said to be deﬁnable in A iff there is a formula (whose free variables are among v1 , . . . , vk ) that deﬁnes it there. EXAMPLE. Assume that we have a part of the language for number theory, speciﬁcally that our language has the parameters ∀, 0, S, +, and ·. Let N be the intended structure: |N| = N, the set of natural numbers. 0N = 0, the number 0. SN , +N , and ·N are S, +, and ·, the functions of successor, addition, and multiplication. In one equation, N = (N; 0, S, +, ·). Some relations on N are deﬁnable in N and some are not. One way to show that some are not deﬁnable is to use the fact that there are uncountably many relations on N but only countably many possible deﬁning formulas. (There is, however, an inherent difﬁculty in giving a speciﬁc example. After all, if something is undeﬁnable, then it is hard to say exactly what it is! Later we will get to see a speciﬁc example, the set of G¨odel numbers of sentences true in N; see Section 3.5.) 1. The ordering relation { m, n | m < n} is deﬁned in N by the formula ∃ v3 v1 + Sv3 = v2 . 2. For any natural number n, {n} is deﬁnable. For example, {2} is deﬁned by the equation v1 = SS0. Because of this we say that n is a deﬁnable element in N. 3. The set of primes is deﬁnable in N. We could use the formula 1 < v1 ∧ ∀ v2 ∀ v3 (v1 = v2 · v3 → v2 = 1 ∨ v3 = 1) if we had parameters 1 and < for 1 and <. But since {1} and < are deﬁnable in N, it is really quite unnecessary to add parameters for them; we can simply use their deﬁnitions instead. Thus the set of primes is deﬁnable by ∃ v3 S0 + Sv3 = v1 ∧ ∀ v2 ∀ v3 (v1 = v2 · v3 → v2 = S0 ∨ v3 = S0). 4. Exponentiation, { m, n, p | p = m n } is also deﬁnable in N. This is by no means obvious; we will give a proof later (in Section 3.8) using the Chinese remainder theorem.

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A Mathematical Introduction to Logic In fact, we will argue later that any decidable relation on N is deﬁnable in N, as is any effectively enumerable relation and a great many others. To some extent the complexity of a deﬁnable relation can be measured by the complexity of the simplest deﬁning formula. This idea will come up again at the end of Section 3.5.

Deﬁnability of a Class of Structures Many a mathematics class, on its ﬁrst day, begins with the instructor saying something like one of the following: 1. “A graph is deﬁned to consist of a nonempty set V together with a set E such that. . . .” 2. “A group is deﬁned to consist of a nonempty set G together with a binary operation ◦ satisfying the axioms. . . .” 3. “An ordered ﬁeld is deﬁned to consist of a nonempty set F together with two binary operations + and · and a binary relation < satisfying the axioms. . . .” 4. “A vector space is deﬁned to consist of a nonempty set V together with a binary operation + and, for each real number r , an operation called scalar multiplication such that. . . .” We want to abstract from this situation. In each case, the objects of study (the graphs, the groups, and so forth) are structures for a suitable language. Moreover, they are required to satisfy a certain set of sentences (referred to as “axioms”). The course in question then studies the models of the set of axioms — or at least some of the models. For a set of sentences, let Mod be the class of all models of , i.e., the class of all structures for the language in which every member of is true. For a single sentence τ we write simply “Mod τ ” instead of “Mod {τ }.” (The reader familiar with axiomatic set theory will notice that Mod , if nonempty, is a proper class; i.e., it is too large to be a set.) A class K of structures for our language is an elementary class (EC) iff K = Mod τ for some sentence τ . K is an elementary class in the wider sense (EC ) iff K = Mod for some set of sentences. (The adjective “elementary” is employed as a synonym for “ﬁrst-order.”) EXAMPLES 1. Assume that the language has equality and the two parameters ∀ and E, where E is a two-place predicate symbol. Then a graph is a structure for this language A = (V ; E A ) consisting of a nonempty set V of objects called vertices (or nodes), and an edge relation E A that is symmetric (if u E A v then v E A u) and irreﬂexive (never v E A v). The axiom stating that the edge relation is symmetric and irreﬂexive can be translated by the sentence

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∀ x(¬ x E x ∧ ∀ y(x E y → y E x)). So the class of all graphs is an elementary class. For directed graphs or digraphs, the assumption of symmetry is dropped. And if one wants to allow “loops” then the assumption of irreﬂexivity is dropped. But perhaps the instructor then explains that the course will study only ﬁnite graphs. Is the class of all ﬁnite graphs an elementary class? No, we will prove later that it is not, not even in the wider sense. 2. Assume that the language has equality and the parameters ∀ and P, where P is a two-place predicate symbol. As before, a structure (A; R) for the language consists of a nonempty set A together with a binary relation R on A. (A; R) is called an ordered set iff R is transitive and satisﬁes the trichotomy condition (which states that for any a and b in A, exactly one of a, b ∈ R, a = b, b, a ∈ R holds). Because these conditions can be translated into a sentence of the formal language, the class of nonempty ordered sets is an elementary class. It is, in fact, Mod τ , where τ is the conjunction of the three sentences ∀ x ∀ y ∀ z(x P y → y P z → x P z); ∀ x ∀ y(x P y ∨ x = y ∨ y P x); ∀ x ∀ y(x P y→ ¬y P x). The next two examples assume that the reader has had some contact with algebra. 3. Assume that the language has = and the parameters ∀ and ◦, where ◦ is a two-place function symbol. The class of all groups is an elementary class, being the class of all models of the conjunction of the group axioms: ∀ x ∀ y ∀ z (x ◦ y) ◦ z = x ◦ (y ◦ z); ∀ x ∀ y ∃ z x ◦ z = y; ∀ x ∀ y ∃ z z ◦ x = y. The class of all inﬁnite groups is EC . To see this, let y, λ2 = ∃ x ∃ y x = λ3 = ∃ x ∃ y ∃ z(x = y∧x= z∧y= z), .... Thus λn translates, “There are at least n things.” Then the group axioms together with {λ2 , λ3 , . . .} form a set for which Mod is exactly the class of inﬁnite groups. We will eventually (in Section 2.6) be able to show that the class of inﬁnite groups is not EC. 4. Assume that the language has equality and the parameters ∀, 0, 1, +, ·. Fields can be regarded as structures for this language.

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A Mathematical Introduction to Logic The class of all ﬁelds is an elementary class. The class of ﬁelds of characteristic zero is EC . It is not EC, a fact which will follow from the compactness theorem for ﬁrst-order logic (Section 2.6 again).

Homomorphisms1 In courses about graphs or groups or vector spaces, one usually encounters the concept of what it means for two of the structures in question, A and B, to be isomorphic: Roughly speaking, there must be a one-to-one correspondence between their universes |A| and |B| that “preserves” the operations and relations. It is then explained that two isomorphic structures, although not identical, must have all the same mathematical properties. We want to deﬁne here the isomorphism concept in a general setting, and to show that two isomorphic structures must satisfy exactly the same sentences. Let A, B be structures for the language. A homomorphism h of A into B is a function h : |A| → |B| with the properties: (a) For each n-place predicate parameter P and each n-tuple a1 , . . . , an of elements of |A|, a1 , . . . , an ∈ P A

iff

h(a1 ), . . . , h(an ) ∈ P B .

(b) For each n-place function symbol f and each such n-tuple, h( f A (a1 , . . . , an )) = f B (h(a1 ), . . . , h(an )). In the case of a constant symbol c this becomes h(cA ) = cB . Conditions (a) and (b) are usually stated: “h preserves the relations and functions.” (It must be admitted that some authors use a weakened version of condition (a); our homomorphisms are their “strong homomorphisms.”) If, in addition, h is one-to-one, it is then called an isomorphism (or an isomorphic embedding) of A into B. If there is an isomorphism of A onto B (i.e., an isomorphism h for which ran h = |B|), then A and B are said to be isomorphic (written A ∼ = B). The reader has quite possibly encountered this concept before in special cases such as structures that are groups or ﬁelds.

1

This topic can be postponed somewhat. But homomorphisms will be used in the proof of the completeness theorem (with equality). And we make use of the isomorphism concept, starting in Section 6 of Chapter 2.

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EXAMPLE. Assume that we have a language with the parameters ∀, +, and ·. Let A be the structure (N; +, ·). We can deﬁne a function h : N → {e, o} by e if n is even, h(n) = o if n is odd. Then h is a homomorphism of A onto B, where |B| = {e, o} and +B , ·B are given by the following tables: +B e o

e e o

o o e

·B e o

e e e

o e o

It can then be veriﬁed that condition (b) of the deﬁnition is satisﬁed. For example, if a and b are both odd numbers, then h(a+b) = e and h(a) +B h(b) = o +B o = e. EXAMPLE. Let P be the set of positive integers, let < P be the usual ordering relation on P, and let < N be the usual ordering relation on N. Then there is an isomorphism h from the structure (P; < P ) onto (N; < N ); we take h(n) = n − 1. Also the identity map I d : P → N is an isomorphism of (P; < P ) into (N; < N ). Because of this last fact, we say that (P; < P ) is a substructure of (N; < N ). More generally consider two structures A and B for the language such that |A| ⊆ |B|. It is clear from the deﬁnition of homomorphism that the identity map from |A| into |B| is an isomorphism of A into B iff (a) P A is the restriction of P B to |A|, for each predicate parameter P; (b) f A is the restriction of f B to |A|, for each function symbol f , and cA = cB for each constant symbol c. If these conditions are met, then A is said to be a substructure of B, and B is an extension of A. For example, in a language with a two-place function symbol +, the structure (Q; + Q ) is a substructure of (C; +C ). Here +C is the addition operation on complex numbers. And + Q , addition on the rationals, is exactly the restriction of +C to the set Q. In this example, the set Q is closed under +C ; that is, the sum of two rational numbers is rational. More generally, whenever A is a substructure of B, then |A| must be closed under f B for every function symbol a ) (where a ∈ |A|n ) is nothing but f A ( a ), which must f . After all, f B ( be some element in |A|. This closure property even holds for the 0-place function symbols; cB must belong to |A| for each constant symbol c.

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A Mathematical Introduction to Logic Conversely, suppose we have a structure B, and let A be any nonempty subset of |B| that is closed under all of B’s functions, as in the preceding paragraph. Then we can make a substructure of B with universe A. In fact there is only one way to do this. The universe is A, each predicate parameter P is assigned the restriction of P B to A, and similarly for the function symbols. As an extreme case, if the language has no function symbols at all (not even constant symbols), then we can make a substructure out of any nonempty subset A of |B|. These are basically algebraic concepts, but the following theorem relates them to the logical concepts of truth and satisfaction. HOMOMORPHISM THEOREM Let h be a homomorphism of A into B, and let s map the set of variables into |A|. (a) For any term t, we have h(s(t)) = h ◦ s(t), where s(t) is computed in A and h ◦ s(t) is computed in B. (b) For any quantiﬁer-free formula α not containing the equality symbol, |=A α[s]

iff

|=B α[h◦s].

(c) If h is one-to-one (i.e., is an isomorphism of A into B), then in part (b) we may delete the restriction “not containing the equality symbol.” (d) If h is a homomorphism of A onto B, then in (b) we may delete the restriction “quantiﬁer-free.” PROOF. Part (a) uses induction on t; see Exercise 13. Note that h ◦ s maps the set of variables into |B|; its extension to the set of all terms is h ◦ s. It is h ◦ s that is here being evaluated at t. (b) For an atomic formula such as Pt, we have |=A Pt[s] ⇔ s(t) ∈ P A ⇔ h(s(t)) ∈ P B since h is a homomorphism ⇔ h ◦ s(t) ∈ P B by (a) ⇔ |=B Pt[h ◦ s]. An inductive argument is then required to handle the connective symbols ¬ and → , but it is completely routine. (c) In any case, |=A u = t[s] ⇔ s(u) = s(t) ⇒ h(s(u)) = h(s(t)) ⇔ h ◦ s(u) = h ◦ s(t) by (a) ⇔ |=B u = t[h ◦ s]. If h is one-to-one, the arrow in the second step can be reversed. (d) We must extend the routine inductive argument of part (b) to include the quantiﬁer step. That is, we must show that if ϕ has

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the property that for every s, |=A ϕ[s] ⇔ |=B ϕ[h ◦ s], then ∀ x ϕ enjoys the same property. We have in any case (as a consequence of the inductive hypothesis on ϕ) the implication |=B ∀ x ϕ[h ◦ s] ⇒ |=A ∀ x ϕ[s]. This is intuitively very plausible; if ϕ is true of everything in the larger set |B|, then a fortiori it is true of everything in the smaller set ran h. The details are, for an element a of |A|, |=B ∀ x ϕ[h ◦ s]⇒ |=B ϕ[(h ◦ s)(x | h(a))] ⇔ |=B ϕ[h ◦ (s(x | a))], ⇔ |=A ϕ[s(x | a)]

the functions being the same by the inductive hypothesis.

Now for the converse, suppose that |=B ∀ x ϕ[h ◦ s], so that |=B ¬ ϕ[(h ◦ s)(x | b)] for some element b in |B|. We need the implication (∗) If for some b in |B|, |=B ¬ ϕ[(h ◦ s)(x | b)], then for some a in |A|, |=B ¬ ϕ[(h ◦ s)(s | h(a))]. For given (∗), we can proceed: |=B ¬ ϕ[(h ◦ s)(x | h(a))] ⇔ |=B ¬ ϕ[h ◦ (s(x | a))], ⇔ |=A ¬ ϕ[s(x | a)]

the functions being the same by the inductive hypothesis

⇒ |=A ∀ x ϕ[s]. If h maps |A| onto |B|, then (∗) is immediate; we take a such that b = h(a). (But there might be other fortunate times when (∗) can be asserted even if h fails to have range |B|.) Two structures A and B for the language are said to be elementarily equivalent (written A ≡ B) iff for any sentence σ , |=A σ ⇔ |=B σ. COROLLARY 22D

Isomorphic structures are elementarily equivalent: A∼ =B ⇒ A≡B

Actually more is true. Isomorphic structures are alike in every “structural” way; not only do they satisfy the same ﬁrst-order sentences, they also satisfy the same second-order (and higher) sentences (i.e., they are secondarily equivalent and more). There are elementarily equivalent structures that are not isomorphic. For example, it can be shown that the structure (R; < R ) consisting of the set of real numbers with its usual ordering relation is elementarily

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A Mathematical Introduction to Logic equivalent to the structure (Q; < Q ) consisting of the set of rational numbers with its ordering (see Section 2.6). But Q is a countable set whereas R is not, so these structures cannot be isomorphic. In Section 2.6 we will see how easy it is to make elementarily equivalent structures of differing cardinalities. EXAMPLE, revisited. We had an isomorphism h from (P; < P ) onto (N; < N ). So in particular, (P; < P ) ≡ (N; < N ); these structures are indistinguishable by ﬁrst-order sentences. We furthermore noted that the identity map was an isomorphic embedding of (P; < P ) into (N; < N ). Hence for a function s: V → P and a quantiﬁer-free ϕ, |=(P;< P ) ϕ[s] ⇔ |=(N;< N ) ϕ[s]. This equivalence may fail if ϕ contains quantiﬁers. For example, |=(P;< P ) ∀ v2 (v1 = v2 → v1 < v2 )[[1]], but

|=(N;< N ) ∀ v2 (v1 = v2 → v1 < v2 )[[1]]. An automorphism of the structure A is an isomorphism of A onto A. The identity function on |A| is trivially an automorphism of A. A may or may not have nontrivial automorphisms. (We say that A is rigid if the identity function is its only automorphism.) As a consequence of the homomorphism theorem, we can show that an automorphism must preserve the deﬁnable relations: COROLLARY 22E Let h be an automorphism of the structure A, and let R be an n-ary relation on |A| deﬁnable in A. Then for any a1 , . . . , an in |A|, a1 , . . . , an ∈ R ⇔ h(a1 ), . . . , h(an ) ∈ R. PROOF. that

Let ϕ be a formula that deﬁnes R in A. We need to know |=A ϕ[[a1 , . . . , an ]] ⇔ |=A ϕ[[h(a1 ), . . . , h(an )]].

But this is immediate from the homomorphism theorem.

This corollary is sometimes useful in showing that a given relation is not deﬁnable. Consider, for example, the structure (R; <) consisting of the set of real numbers with its usual ordering. An automorphism of this structure is simply a function h from R onto R that is strictly increasing: a < b ⇔ h(a) < h(b).

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One such automorphism is the function h for which h(a) = a 3 . Since this function maps points outside of N into N, the set N is not deﬁnable in this structure. Another example is provided by elementary algebra books, which sometimes explain that the length of a vector in the plane cannot be deﬁned in terms of vector addition and scalar multiplication. For the map that takes a vector x into the vector 2x is an automorphism of the plane with respect to vector addition and scalar multiplication, but it is not length-preserving. From our viewpoint, the structure in question, (E; +, fr )r ∈R , has for its universe the plane E, has the binary operation + of vector addition, and has (for each r in the set R) the unary operation fr of scalar multiplication by r . (Thus the language in question has a one-place function symbol for each real number.) The doubling map described above is an automorphism of this structure. But it does not preserve the set of unit vectors, {x | x ∈ E and x has length 1}. So this set cannot be deﬁnable in the structure. (Incidentally, the homomorphisms of vector spaces are called linear transformations.)

Exercises 1. Show that (a) ; α |= ϕ iff |= (α → ϕ); and (b) ϕ |==| ψ iff |= (ϕ ↔ ψ). 2. Show that no one of the following sentences is logically implied by the other two. (This is done by giving a structure in which the sentence in question is false, while the other two are true.) (a) ∀ x ∀ y ∀ z(P x y → P yz → P x z). Recall that by our convention α → β → γ is α → (β → γ ). (b) ∀ x ∀ y(P x y → P yx → x = y). (c) ∀ x ∃ y P x y → ∃ y ∀ x P x y. 3. Show that {∀ x(α → β), ∀ x α} |= ∀ x β. 4. Show that if x does not occur free in α, then α |= ∀ x α. 5. Show that the formula x = y → P z f x → P z f y (where f is a one-place function symbol and P is a two-place predicate symbol) is valid. 6. Show that a formula θ is valid iff ∀ x θ is valid. 7. Restate the deﬁnition of “A satisﬁes ϕ with s” in the way described

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A Mathematical Introduction to Logic on page 84. That is, deﬁne by recursion a function h such that A satisﬁes ϕ with s iff s ∈ h(ϕ). 8. Assume that is a set of sentences such that for any sentence τ , either |= τ or |= ¬ τ . Assume that A is a model of . Show that for any sentence τ , we have |=A τ iff |= τ . 9. Assume that the language has equality and a two-place predicate symbol P. For each of the following conditions, ﬁnd a sentence σ such that the structure A is a model of σ iff the condition is met. (a) |A| has exactly two members. (b) P A is a function from |A| into |A|. (A function is a single-valued relation, as in Chapter 0. For f to be a function from A into B, the domain of f must be all of A; the range of f is a subset, not necessarily proper, of B.) (c) P A is a permutation of |A|; i.e., P A is a one-to-one function with domain and range equal to |A|. 10. Show that |=A ∀ v2 Qv1 v2 [[cA ]]

iff

|=A ∀ v2 Qcv2 .

Here Q is a two-place predicate symbol and c is a constant symbol. 11. For each of the following relations, give a formula which deﬁnes it in (N; +, ·). (The language is assumed to have equality and the parameters ∀, +, and ·). (a) {0}. (b) {1}. (c) { m, n | n is the successor of m in N }. (d) { m, n | m < n in N}. Digression: This is merely the tip of the iceberg. A relation on N is said to be arithmetical if it is deﬁnable in this structure. All decidable relations are arithmetical, as are many others. The arithmetical relations can be arranged in a hierarchy; see Section 3.5. 12. Let R be the structure (R; +, ·). (The language is assumed to have equality and the parameters ∀, +, and ·. R is the structure whose universe is the set R of real numbers and such that +R and ·R are the usual addition and multiplication operations.) (a) Give a formula that deﬁnes in R the interval [0, ∞). (b) Give a formula that deﬁnes in R the set {2}. ∗ (c) Show that any ﬁnite union of intervals, the endpoints of which are algebraic, is deﬁnable in R. (The converse is also true; these are the only deﬁnable sets in the structure. But we will not prove this fact.) 13. Prove part (a) of the homomorphism theorem.

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14. What subsets of the real line R are deﬁnable in (R; <)? What subsets of the plane R × R are deﬁnable in (R; <)? Remarks: The nice thing about (R; <) is that its automorphisms are exactly the order-preserving maps from R onto itself. But stop after the binary relations. There are 213 deﬁnable ternary relations, so you do not want to catalog all of them. 15. Show that the addition relation, { m, n, p | p = m + n}, is not deﬁnable in (N; ·). Suggestion: Consider an automorphism of (N; ·) that switches two primes. Digression: Algebraically, the structure of the natural numbers with multiplication is nothing but the free Abelian semigroup with ℵ0 generators (viz. the primes), together with a zero element. There is no way you could deﬁne addition here. If you could deﬁne addition, then you could deﬁne ordering (by Exercise 11 and the natural transitivity statement). But one generator looks just like another. That is, there are 2ℵ0 automorphisms — simply permute the primes. None of them is order-preserving except the identity. 16. Give a sentence having models of size 2n for every positive integer n, but no ﬁnite models of odd size. (Here the language should include equality and will have whatever parameters you choose.) Suggestion: One method is to make a sentence that says, “Everything is either red or blue, and f is a color-reversing permutation.” Remark: Given a sentence σ , it might have some ﬁnite models (i.e., models with ﬁnite universes). Deﬁne the spectrum of σ to be the set of positive integers n such that σ has a model of size n. This exercise shows that the set of even numbers is a spectrum. For example if σ is the conjunction of the ﬁeld axioms (there are only ﬁnitely many, so we can take their conjunction), then its spectrum is the set of powers of primes. This fact is proved in any course on ﬁnite ﬁelds. The spectrum of ¬ σ , by contrast, is the set of all positive integers (non-ﬁelds come in all sizes). G¨unter Asser in 1955 raised the question: Is the complement of every spectrum a spectrum? Once you realize that simply taking a negation does not work (cf. the preceding paragraph), you see that this is a nontrivial question. In fact the problem, known as the spectrum problem, is still open. But modern work has tied it to another open problem, whether or not co-NP = NP. 17. (a) Consider a language with equality whose only parameter (aside from ∀) is a two-place predicate symbol P. Show that if A is ﬁnite and A ≡ B, then A is isomorphic to B. Suggestion: Suppose the universe of A has size n. Make a single sentence σ of the form ∃ v1 · · · ∃ vn θ that describes A “completely.” That is, on the one hand, σ must be true in A. And on the other hand, any model of σ must be exactly like (i.e., isomorphic to) A.

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(b) Show that the result of part (a) holds regardless of what parameters the language contains. 18. A universal (∀1 ) formula is one of the form ∀ x1 · · · ∀ xn θ , where θ is quantiﬁer-free. An existential (∃1 ) formula is of the dual form ∃ x1 · · · ∃ xn θ . Let A be a substructure of B, and let s : V → |A|. (a) Show that if |=A ψ[s] and ψ is existential, then |=B ψ[s]. And if |=B ϕ[s] and ϕ is universal, then |=A ϕ[s]. (b) Conclude that the sentence ∃ x P x is not logically equivalent to any universal sentence, nor ∀ x P x to any existential sentence. Remark: Part (a) says (when ϕ is a sentence) that any universal sentence is “preserved under substructures.” Being universal is a syntactic property — it has to do with the string of symbols. In contrast, being preserved under substructures is a semantic property — it has to do with satisfaction in structures. But this semantic property captures the syntactic property up to logical equivalence (which is all one could ask for). That is, if σ is a sentence that is always preserved under substructures, then σ is logically equivalent to a universal sentence. (This fact is due to Ło´s and Tarski.) 19. An ∃2 formula is one of the form ∃ x1 · · · ∃ xn θ, where θ is universal. (a) Show that if an ∃2 sentence in a language not containing function symbols (not even constant symbols) is true in A, then it is true in some ﬁnite substructure of A. (b) Conclude that ∀ x ∃ y P x y is not logically equivalent to any ∃2 sentence. 20. Assume the language has equality and a two-place predicate symbol P. Consider the two structures (N; <) and (R; <) for the language. (a) Find a sentence true in one structure and false in the other. ∗ (b) Show that any ∃2 sentence (as deﬁned in the preceding exercise) true in (R; <) is also true in (N; <). Suggestion: First, for any ﬁnite set of real numbers, there is an automorphism of (R; <) taking those real numbers to natural numbers. Secondly, by Exercise 18, universal formulas are preserved under substructures. 21. We could consider enriching the language by the addition of a new quantiﬁer. The formula ∃!x α (read “there exists a unique x such that α”) is to be satisﬁed in A by s iff there is one and only one a ∈ |A| such that |=A α[s(x | a)]. Assume that the language has the equality symbol and show that this apparent enrichment comes to naught, in the sense that we can ﬁnd an ordinary formula logically equivalent to ∃!x α. 22. Assume that A is a structure and h is a function with ran h = |A|.

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Show that there is a structure B such that h is a homomorphism of B onto A. Suggestion: We need to take |B| = dom h. In general, the axiom of choice will be needed to deﬁne the functions in B, unless h is one-to-one. Remark: The result yields an “upward L¨owenheim–Skolem theorem without equality” (cf. Section 2.6). That is, any structure A has an extension to a structure B of any higher cardinality such that A and B are elementarily equivalent, except for equality. There is nothing deep about this. Not until you add equality. 23. Let A be a structure and g a one-to-one function with dom g = |A|. Show that there is a unique structure B such that g is an isomorphism of A onto B. 24. Let h be an isomorphic embedding of A into B. Show that there is a structure C isomorphic to B such that A is a substructure of C. Suggestion: Let g be a one-to-one function with domain |B| such that g(h(a)) = a for a ∈ |A|. Form C such that g is an isomorphism of B onto C. Remark: The result stated in this exercise should not seem surprising. On the contrary, it is one of those statements that is obvious until you have to prove it. It says that if you can embed A isomorphically into B, then for all practical purposes you can pretend A is a substructure of B. 25. Consider a ﬁxed structure A. Expand the language by adding a new constant symbol ca for each a ∈ |A|. Let A+ be the structure for this expanded language that agrees with A on the original parameters and that assigns to ca the point a. A relation R on |A| is said to be deﬁnable from points in A iff R is deﬁnable in A+ . (This differs from ordinary deﬁnability only in that we now have parameters in the language for members of |A|.) Let R = (R; <, +, ·). (a) Show that if A is a subset of R consisting of the union of ﬁnitely many intervals, then A is deﬁnable from points in R (cf. Exercise 12). (b) Assume that A ≡ R. Show that any subset of |A| that is nonempty, bounded (in the ordering

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A Mathematical Introduction to Logic order property. See Example 4 in Section 4.1 for its second-order statement. The ﬁrst-order “image” of completeness is given by the schema obtained from that second-order statement by replacing X by a ﬁrst-order formula ϕ. The resulting schema (i.e., the set of sentences you get by letting ϕ vary and taking universal closure) says that the least-upper-bound property holds for the sets that are deﬁnable from points. Ordered ﬁelds satisfying those sentences are called “real closed-ordered ﬁelds.” The surprising fact is that such ﬁelds were not invented by logicians. They were previously studied by algebraists and you can read about them in van der Waerden’s Modern Algebra book (volume I). Of course, he uses a characterization of them that does not involve logic. What Tarski showed is that any real closed-ordered ﬁeld is elementarily equivalent to the ﬁeld of real numbers. From this it follows that the theory of the real-ordered ﬁeld is decidable. 26. (a) Consider a ﬁxed structure A and deﬁne its elementary type to be the class of structures elementarily equivalent to A. Show that this class is EC . Suggestion: Show it is Mod Th A. (b) Call a class K of structures elementarily closed or ECL if whenever a structure belongs to K then all elementarily equivalent structures also belong. Show that any such class is a union of EC classes. (A class that is a union of EC classes is said to be an EC class; this notation is derived from topology.) (c) Conversely, show that any class that is the union of EC classes is elementarily closed. 27. Assume that the parameters of the language are ∀ and a two-place predicate symbol P. List all of the non-isomorphic structures of size 2. That is, give a list of structures (where the universe of each has size 2) such that any structure of size 2 is isomorphic to exactly one structure on the list. 28. For each of the following pairs of structures, show that they are not elementarily equivalent, by giving a sentence true in one and false in the other. (The language here contains ∀ and a two-place function symbol ◦.) (a) (R; ×) and (R∗ ; ×∗ ), where × is the usual multiplication operation on the real numbers, R∗ is the set of non-zero reals, and ×∗ is × restricted to the non-zero reals. (b) (N; +) and (P; +∗ ), where P is the set of positive integers, and +∗ is usual addition operation restricted to P. (c) Better yet, for each of the four structures of parts (a) and (b), give a sentence true in that structure and false in the other three.

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SECTION 2.3 A Parsing Algorithm1 As in sentential logic, we need to know that we can decompose formulas (and terms) in a unique way, discovering how they are built up. The uniqueness is necessary to justify our deﬁnitions by recursion, such as the deﬁnition of satisfaction in the preceding section. For terms we used Polish notation; for formulas we relied on parentheses. Accordingly, we ﬁrst consider a decomposition procedure for terms, showing unique readability. And then we extend the methods to cover the formulas. Recall that the terms are built up from the variables and constant symbols by operations corresponding to the function symbols. We now deﬁne a function K on the symbols involved such that for a symbol s, K (s) = 1 − n, where n is the number of terms that must follow s to obtain a term: K (x) = 1 − 0 = 1 for a variable x; K (c) = 1 − 0 = 1 for a constant symbol c; K( f ) = 1 − n for an n-place function symbol f. We then extend K to the set of expressions using these symbols by deﬁning K (s1 s2 · · · sn ) = K (s1 ) + K (s2 ) + · · · + K (sn ). Since no symbol is a ﬁnite sequence of others, this deﬁnition is unambiguous. LEMMA 23A For any term t, K (t) = 1. PROOF. Use induction on t. The inductive step, for an n-place function symbol f , is K ( f t1 · · · tn ) = (1 − n) + (1 + · · · + 1) = 1. n times

In fact, K was chosen to be the unique function on these symbols for which Lemma 23A holds. It follows from this lemma that if ε is a concatenation of m terms, then K (ε) = m. By a terminal segment of a string s1 , . . . , sn of symbols we mean a sequence of the form sk , sk+1 , . . . , sn , where 1 ≤ k ≤ n. LEMMA 23B Any terminal segment of a term is a concatenation of one or more terms.

1

This section may be omitted by a reader willing to accept the meaningfulness of our many deﬁnitions by recursion.

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A Mathematical Introduction to Logic PROOF. We use induction on the term. For a one-symbol term (i.e., a variable or a constant symbol) the conclusion follows trivially. For a term f t1 · · · tn , any terminal segment (other than the term itself) must equal tk tk+1 · · · tn , where k ≤ n and tk is a terminal segment of tk . By the inductive hypothesis tk is a concatenation of, say, m terms, where m ≥ 1. So altogether we have m + (n − k) terms. COROLLARY 23C No proper initial segment of a term is itself a term. If t1 is a proper initial segment of a term t, then K (t1 ) < 1. PROOF. Suppose a term t is divided into a proper initial segment t1 and a terminal segment t2 . Then 1 = K (t) = K (t1 ) + K (t2 ), and by Lemma 23B, K (t2 ) ≥ 1. Hence K (t1 ) < 1 and t1 cannot be a term.

Parsing Terms We want an algorithm that, given an expression, will determine whether or not the expression is a legal term, and if it is, will construct the unique tree showing how the term is built up. Assume then that we are given an expression. We will construct a tree, with the given expression at the top (i.e., the root). Initially it is the only vertex in the tree, but as the procedure progresses, the tree will grow downward. The algorithm consist of the following two steps. 1. If each minimal vertex (at the bottom) has a single symbol (which must∗ be a variable or a constant symbol), then the procedure is completed. (The given expression is indeed a term, and we have constructed its tree.) Otherwise, select a minimal vertex having an expression with two or more symbols. We examine that expression. 2. The ﬁrst symbol must* be an n-place function symbol, say f , where n > 0. We extend the tree downward by creating n new vertices below the present one. Scan the expression after the f , until ﬁrst reaching a string t (of variables, constant symbols, and function symbols) with K (t) = 1.∗∗ Then t is the expression that goes at the leftmost unlabeled new vertex. Repeat with the remainder of the expression, until all n new vertices have been labeled, and the expression has been exhausted.** Return to step 1.

∗ ∗∗

If not, then the expression here is not a term. We reject the given expression as a non-term, and halt. If the end of the expression is reached before ﬁnding such an t, then the expression here is not a term. We reject the given expression as a non-term, and halt.

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As in Section 1.3, the crucial point is that the tree could not have been made differently. In step 2, we selected the ﬁrst string t we found with K (t) = 1. We could not have used less than t (because we needed K (t) = 1 by Lemma 23A). We could not have used more than t (because that longer string would have the proper initial segment t with K (t) = 1, contradicting Corollary 23C). The choice of t was the only possible choice. When the algorithm halts, either it has rejected the given expression as a non-term, or it has constructed the one tree demonstrating that the given expression is a legal tree. We can rephrase the uniqueness in the terminology of Section 1.4 as follows: UNIQUE READABILITY THEOREM FOR TERMS The set of terms is freely generated from the set of variables and constant symbols by the F f operations.

g, then ran F f is disjoint from PROOF. First, it is clear that iff f = ran Fg ; this requires checking only the ﬁrst symbol. Furthermore, both ranges are disjoint from the set of variables and constant symbols. It remains only to show that F f , when restricted to terms, is one-to-one. Suppose, for a two-place f , we have f t1 t2 = f t3 t4 . By deleting the ﬁrst symbol we are left with t1 t2 = t3 t4 .

t3 , then one would be a proper initial segment of the other, If t1 = which is impossible for terms by Corollary 23C. So t1 = t3 , and we are then left with t2 = t4 .

Parsing Formulas To extend this argument to formulas, we now deﬁne K on the other symbols: K (() = −1; K ()) = 1; K (∀) = −1; K (¬) = 0; K (→) = −1; K (P) = 1 − n K (=) = −1.

for an n-place predicate symbol P;

The idea behind the deﬁnition is again that K (s) should be 1 − n, where n is the number of things (right parentheses, terms, or formulas) required to go after s. We extend K as usual to the set of all expressions: K (s1 · · · sn ) = K (s1 ) + · · · + K (sn ).

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A Mathematical Introduction to Logic LEMMA 23D For any wff α, K (α) = 1. PROOF.

Another straightforward induction.

LEMMA 23E For any proper initial segment α of a wff α, K (α ) < 1. PROOF.

Use induction on α. The details are left to Exercise 1.

COROLLARY 23F formula.

No proper initial segment of a formula is itself a

Armed with these facts, we can proceed as in Section 1.3. Instead of sentence symbols at the minimal vertices, we now have atomic formulas (which are distinguished by having ﬁrst an n-place predicate symbol, followed by n terms). A wff that is non-atomic must begin either with ∀ vi or with (. In the former case we have one new vertex; in the latter case we need to examine the next symbol to see if it is ¬. If not, then we can either count parentheses or use the K function — both methods work — to ﬁnd the correct split. Again, the uniqueness can be phrased in the terminology of Section 1.4 as follows: UNIQUE READABILITY THEOREM FOR FORMULAS The set of wffs is freely generated from the set of atomic formulas by the operations E ¬ , E → , and Qi (i = 1, 2, . . .). PROOF. The unary operations E¬ and Qi are obviously one-to-one. As in Section 1.4, we can show that the restriction of E → to wffs is one-to-one. The disjointness half of the theorem follows from the ad hoc observations: 1. ran E¬ , ran Qi , ran Q j , and the set of atomic formulas are pairwise disjoint, for i =

j. (Just look at the ﬁrst two symbols.) 2. ran E → , ran Qi , ran Q j , and the set of atomic formulas are similarly pairwise disjoint, for i =

j. 3. For a wff β, ( ¬ α) =

(β → γ ), because no wff begins with ¬. Hence ran E¬ is disjoint from the range of the restriction of E → to wffs.

Exercises 1. Show that for a proper initial segment α of a wff α, we have K (α ) < 1. 2. Let ε be an expression consisting of variables, constant symbols, and function symbols. Show that ε is a term iff K (ε) = 1 and for every terminal segment ε of ε we have K (ε ) > 0. Suggestion: Prove the

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stronger result that if K (ε ) > 0 for every terminal segment ε of ε, then ε is a concatenation of K (ε) terms. (This algorithm is due to J´askowski.)

SECTION 2.4 A Deductive Calculus Suppose that |= τ . What methods of proof might be required to demonstrate that fact? Is there necessarily a proof at all? Such questions lead immediately to considerations of what constitutes a proof. A proof is an argument that you give to someone else and that completely convinces that person of the correctness of your assertion (in this case, that |= τ ). Thus a proof should be ﬁnitely long, as you cannot give all of an inﬁnite object to another person. If the set of hypotheses is inﬁnite, they cannot all be used. But the compactness theorem for ﬁrst-order logic (which we will prove in Section 2.5 using the deductive calculus of this section) will ensure the existence of a ﬁnite 0 ⊆ such that 0 |= τ . Another essential feature of a proof (besides its being ﬁnite in length) is that it must be possible for someone else (if that person is to be convinced by it) to check the proof to ascertain that it contains no fallacies. This check must be effective; it must be the sort of thing that can be carried out without brilliant ﬂashes of insight on the part of the checker. In particular, the set of proofs from the empty set of hypotheses (i.e., proofs that |= τ ) should be decidable. This implies that the set of formulas provable without hypotheses must be effectively enumerable. For one could in principle enumerate provable sentences by generating all strings of symbols and sorting out the proofs from the nonproofs. When a proof is discovered, its last line is entered on the output list. (This issue will be examined more carefully at the end of Section 2.5.) But here again there is a theorem (the enumerability theorem, proved in Section 2.5) stating that, under reasonable conditions, the validities — the set of valid formulas — is indeed effectively enumerable. Thus the compactness theorem and the enumerability theorem are necessary conditions for satisfactory proofs of logical implication always to exist. Conversely, we claim that these two theorems are sufﬁcient for proofs (of some sort) to exist. For suppose that |= τ . By the compactness theorem, then, there is a ﬁnite set {σ0 , . . . , σn } ⊆ that logically implies τ . Then σ0 → · · · → σn → τ is valid (Exercise 1, Section 2.2). So to demonstrate conclusively that |= τ one need only carry out a ﬁnite number of steps in the enumeration of the validities

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A Mathematical Introduction to Logic until σ0 → · · · → σn → τ appears, and then verify that each σi ∈ . (This should be compared with the complex procedure suggested by the original deﬁnition of logical implication, discussed in Section 2.2.) The record of the enumeration procedure that produced σ0 → · · · → σn → τ can then be regarded as a proof that |= τ . As a proof, it should be acceptable to anyone who accepts the correctness of your procedure for enumerating validities. Against the foregoing general (and slightly vague) discussion, the outline of this section can be described as follows: We will introduce formal proofs but we will call them deductions, to avoid confusion with our English-language proofs. These will mirror (in our model of deductive thought) the proofs made by the working mathematician to convince his or her colleagues of certain truths. Then in Section 2.5 we will show that whenever |= τ , there is a deduction of τ from (and only then). This will, as is suggested by the foregoing discussion, yield proofs of the compactness theorem and the enumerability theorem. And in the process we will get to see what methods of deduction are adequate to demonstrate that a given sentence, is, in fact, logically implied by certain others. In other words, our goal is to produce a mathematically precise concept of deduction that, in the context of ﬁrst-order logic, is adequate and correct.

Formal Deductions We will shortly select an inﬁnite set of formulas to be called logical axioms. And we will have a rule of inference, which will enable us to obtain a new formula from certain others. Then for a set of formulas, the theorems of will be the formulas which can be obtained from ∪ by use of the rule of inference (some ﬁnite number of times). If ϕ is a theorem of (written ! ϕ), then a sequence of formulas that records (as explained below) how ϕ was obtained from ∪ with the rule of inference will be called a deduction of ϕ from . The choice of and the choice of the rule (or rules) of inference are far from unique. In this section we are presenting one deductive calculus for ﬁrst-order logic, chosen from the array of possible calculi. (For example, one can have = ∅ by using many rules of inference. We will take the opposite extreme; our set will be inﬁnite but we will have only one rule of inference.) Our one rule of inference is traditionally known as modus ponens. It is usually stated: From the formulas α and α → β we may infer β: α, α → β . β Thus the theorems of a set are the formulas obtainable from ∪ by use of modus ponens some ﬁnite number of times.

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DEFINITION. A deduction of ϕ from is a ﬁnite sequence α0 , . . . , αn of formulas such that αn is ϕ and for each k ≤ n, either (a) αk is in ∪ , or (b) αk is obtained by modus ponens from two earlier formulas in the sequence; that is, for some i and j less than k, α j is αi → αk . If such a deduction exists, we say that ϕ is deducible from , or that ϕ is a theorem of , and we write ! ϕ. There is another viewpoint that is useful here: A deduction of ϕ from can be viewed as a construction sequence, showing how ϕ can be obtained from the set ∪ by applying modus ponens zero or more times. (We hesitate to say that ϕ is built “up” from ∪ . Unlike the formula-building operations that produce longer formulas from shorter ones, modus ponens can produce shorter formulas from longer ones.) That is, the set of theorems of is exactly the set of formulas that are obtainable from the “base” set ∪ by applying modus ponens. The situation here differs from the one discussed in Section 1.4 in two ways, one unimportant and one important. The unimportant difference is that we get the set of theorems by closing under the “partially deﬁned” operation of modus ponens, whose domain consists only of pairs of formulas of the form α, α → β (in contrast to the “totally deﬁned” formula-building operations). The more important difference is that the set of theorems is not freely generated from ∪ by modus ponens. This reﬂects the fact that a theorem never has a unique deduction. In Sections 1.3 and 2.3, we were concerned to show that for any wff ϕ, there was a unique tree (which we could effectively calculate) showing how ϕ was built up by use of formula-building operations. Now the tree is by no means unique, and calculating one such tree is a very different matter from before. Nevertheless, this viewpoint does yield the following induction principle. We say that a set S of formulas is closed under modus ponens if whenever both α ∈ S and (α → β) ∈ S then also β ∈ S. INDUCTION PRINCIPLE Suppose that S is a set of wffs that includes ∪ and is closed under modus ponens. Then S contains every theorem of . For example, if the formulas β, γ , and γ → β → α are all in ∪ , then ! α, as is evidenced by the tree which displays how α was

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A Mathematical Introduction to Logic obtained. Although it is tempting (and in some ways more elegant) to deﬁne a deduction to be such a tree, it will be simpler to take deductions to be the linear sequences obtained by squashing such trees into straight lines. Now at last we give the set of logical axioms. These are arranged in six groups. Say that a wff ϕ is a generalization of ψ iff for some n ≥ 0 and some variables x1 , . . . , xn , ϕ = ∀ x1 · · · ∀ xn ψ. We include the case n = 0; any wff is a generalization of itself. The logical axioms are then all generalizations of wffs of the following forms, where x and y are variables and α and β are wffs: 1. 2. 3. 4.

Tautologies; ∀ x α → αtx , where t is substitutable for x in α; ∀ x(α → β) → ( ∀ x α → ∀ x β); α → ∀ x α, where x does not occur free in α.

And if the language includes equality, then we add 5. x = x; 6. x = y → (α → α ), where α is atomic and α is obtained from α by replacing x in zero or more (but not necessarily all) places by y. For the most part groups 3–6 are self-explanatory; we will see various examples later. Groups 1 and 2 require explanation. But ﬁrst we should admit that the above list of logical axioms may not appear very natural. Later it will be possible to see where each of the six groups originated.

Substitution In axiom group 2 we ﬁnd ∀ x α → αtx . Here αtx is the expression obtained from the formula α by replacing the variable x, wherever it occurs free in α, by the term t. This concept can also be (and for us ofﬁcially is) deﬁned by recursion: 1. For atomic α, αtx is the expression obtained from α by replacing the variable x by t. (This is elaborated upon in Exercise 1. Note that αtx is itself a formula.) 2. (¬ α)tx = (¬ αtx ). 3. (α → β)tx =(αtx → βtx ). ∀yα if x = y, 4. (∀ y α)tx = ∀ y(αtx ) if x =

y.

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EXAMPLES 1. ϕxx = ϕ. 2. (Qx → ∀ x P x)xy = (Qy → ∀ x P x). 3. If α is ¬ ∀ y x = y, then ∀ x α → αzx is ∀ x ¬ ∀ y x = y → ¬ ∀ y z = y. 4. For α as in 3, ∀ x α → α yx is ∀ x ¬ ∀ y x = y → ¬ ∀ y y = y. The last example above illustrates a hazard which must be guarded against. On the whole, ∀ x α → αtx seems like a plausible enough axiom. (“If α is true of everything, then it should be true of t.”) But in Example 4, we have a sentence of the form ∀ x α → αtx , which is nearly always false. The antecedent, ∀ x ¬ ∀ y x = y, is true in any structure whose universe contains two or more elements. But the consequent, ¬ ∀ y y = y, is false in any structure. So something has gone wrong. The problem is that when y was substituted for x, it was immediately “captured” by the ∀ y quantiﬁer. We must impose a restriction on axiom group 2 that will preclude this sort of quantiﬁer capture. Informally, we can say that a term t is not substitutable for x in α if there is some variable y in t that is captured by a ∀ y quantiﬁer in αtx . The real deﬁnition is given below by recursion. (Since the concept will be used later in proofs by induction, a recursive deﬁnition is actually the most usable variety.) Let x be a variable, t a term. We deﬁne the phrase “t is substitutable for x in α” as follows: 1. For atomic α, t is always substitutable for x in α. (There are no quantiﬁers in α, so no capture could occur.) 2. t is substitutable for x in (¬ α) iff it is substitutable for x in α. t is substitutable for x in (α → β) iff it is substitutable for x in both α and β. 3. t is substitutable for x in ∀ y α iff either (a) x does not occur free in ∀ y α, or (b) y does not occur in t and t is substitutable for x in α. (The point here is to be sure that nothing in t will be captured by the ∀ y preﬁx and that nothing has gone wrong inside α earlier.) For example, x is always substitutable for itself in any formula. If t contains no variables that occur in α, then t is substitutable for x in α. The reader is cautioned not to be confused about the choice of words. Even if t is not substitutable for x in α, still αtx is obtained from α by replacing x wherever it occurs free by t. Thus in forming αtx , we carry out the indicated substitution even if a prudent person would think it unwise to do so.

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A Mathematical Introduction to Logic Axiom group 2 consists of all generalizations of formulas of the form ∀ x α → αtx , where the term t is substitutable for the variable x in the formula α. For example, ∀ v3 ( ∀ v1 (Av1 → ∀ v2 Av2 ) → (Av2 → ∀ v2 Av2 )) is in axiom group 2. Here x is v1 , α is Av1 → ∀ v2 Av2 , and t is v2 . On the other hand, ∀ v1 ∀ v2 Bv1 v2 → ∀ v2 Bv2 v2 is not in axiom group 2, since v2 is not substitutable for v1 in ∀ v2 Bv1 v2 .

Tautologies Axiom group 1 consists of generalizations of formulas to be called tautologies. These are the wffs obtainable from tautologies of sentential logic (having only the connectives ¬ and →) by replacing each sentence symbol by a wff of the ﬁrst-order language. For example, ∀ x[( ∀ y ¬ P y → ¬ P x) → (P x → ¬ ∀ y ¬ P y)] belongs to axiom group 1. It is a generalization of the formula in square brackets, which is obtained from a contraposition tautology (A → ¬ B) → (B → ¬ A) by replacing A by ∀ y ¬ P y and B by P x. There is another, more direct, way of looking at axiom group 1. Divide the wffs into two groups: 1. The prime formulas are the atomic formulas and those of the form ∀ x α. 2. The nonprime formulas are the others, i.e., those of the form ¬ α or α → β. Thus any formula is built up from prime formulas by the operations E¬ and E → . Now go back to sentential logic, but take the sentence symbols to be the prime formulas of our ﬁrst-order language. Then any tautology of sentential logic (that uses only the connectives ¬, →) is in axiom group 1. There is no need to replace sentence symbols here by ﬁrst-order wffs; they already are ﬁrst-order wffs. Conversely, anything in axiom group 1 is a generalization of a tautology of sentential logic. (The proof of this uses Exercise 8 of Section 1.2.) EXAMPLE, revisited. ( ∀ y ¬ P y → ¬ P x) → (P x → ¬ ∀ y ¬ P y).

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This has two sentence symbols (prime formulas), ∀ y ¬ P y and P x. So its truth table has four lines: ( ∀ y ¬ Py T T F F

→ ¬ P x) → (P x F FT T T T T F T F T FT T T T T F T F

→ ¬ ∀ y ¬ P y) F FT T FT T T F T T F

From the table we see that it is indeed a tautology. On the other hand, neither ∀ x(P x → P x) nor ∀ x P x → P x is a tautology. One remark: We have not assumed that our ﬁrst-order language has only countably many formulas. So we are potentially employing an extension of Chapter 1 to the case of an uncountable set of sentence symbols. A second remark: Taking all tautologies as logical axioms is overkill. We could get by with much less, at the expense of lengthening deductions. On the one hand, the tautologies form a nice decidable set (the decidability will be important for the enumerability theorem in Section 2.5). On the other hand, there is no known fast decision procedure for tautologies, as noted in Section 1.7. One option would be to cut axiom group 1 down to a set of tautologies for which we do know decision procedures that are fast (the technical term is “polynomial-time decidable”). The other tautologies would then be obtainable from these by use of modus ponens. A third remark: Now that ﬁrst-order formulas are also wffs of sentential logic, we can apply concepts from both Chapters 1 and 2 to them. If tautologically implies ϕ, then it follows that also logically implies ϕ. (See Exercise 3.) But the converse fails. For example, ∀ x P x logically implies Pc. But ∀ x P x does not tautologically imply Pc, as ∀ x P x and Pc are two different sentence symbols. THEOREM 24B

! ϕ iff ∪ tautologically implies ϕ.

PROOF. (⇒): This depends on the obvious fact that {α, α → β} tautologically implies β. Suppose that we have a truth assignment v that satisﬁes every member of ∪ . By induction we can see that v satisﬁes any theorem of . The inductive step uses exactly the above-mentioned obvious fact. (⇐): Assume that ∪ tautologically implies ϕ. Then by the corollary to the compactness theorem (for sentential logic), there is a ﬁnite subset {γ1 , . . . , γm , λ1 , . . . , λn } that tautologically implies ϕ. Consequently, γ1 → · · · → γ m → λ 1 → · · · λn → ϕ

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A Mathematical Introduction to Logic is a tautology (cf. Exercise 4 of Section 1.2) and hence is in . By applying modus ponens m + n times to this tautology and to {γ1 , . . . , γm , λ1 , . . . , λn } we obtain ϕ. (The above proof is related to Exercise 7 in Section 1.7. It uses sentential compactness for a possibly uncountable language.)

Deductions and Metatheorems We now have completed the description of the set of logical axioms. The set of theorems of a set is the set built up from ∪ by modus ponens. For example, ! P x → ∃ y P y. (Here = ∅; we write “! α” in place of “∅ ! α.”) The formula P x → ∃ y P y can be obtained by applying modus ponens (once) to two members of , as displayed by the pedigree tree:

We get a deduction of P x → ∃ y P y (from ∅) by compressing this tree into a linear three-element sequence. As a second example, we can obtain a generalization of the formula in the ﬁrst example: ! ∀ x(P x → ∃ y P y). This fact is evidenced by the following tree, which displays the construction of ∀ x(P x → ∃ y P y) from by modus ponens:

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Again we can compress the tree into a deduction. In these examples the pedigree trees may seem to have been pulled out of the air. But we will shortly develop techniques for generating such trees in a somewhat systematic manner. These techniques will rely heavily on the generalization theorem and the deduction theorem below. Notice that we use the word “theorem” on two different levels. We say that α is a theorem of if ! α. We also make numerous statements in English, each called a theorem, such as the one below. It seems unlikely that any confusion will arise. The English statements could have been labeled metatheorems to emphasize that they are results about deductions and theorems. The generalization theorem reﬂects our informal feeling that if we can prove x without any special assumptions about x, we then are entitled to say that “since x was arbitrary, we have ∀ x x .” GENERALIZATION THEOREM If ! ϕ and x do not occur free in any formula in , then ! ∀ x ϕ. PROOF. Consider a ﬁxed set and a variable x not free in . We will show by induction that for any theorem ϕ of , we have ! ∀ x ϕ. For this it sufﬁces (by the induction principle) to show that the set {ϕ | ! ∀ x ϕ} includes ∪ and is closed under modus ponens. Notice that x can occur free in ϕ. Case 1: ϕ is a logical axiom. Then ∀ x ϕ is also a logical axiom. And so ! ∀ x ϕ. Case 2: ϕ ∈ . Then x does not occur free in ϕ. Hence ϕ →∀x ϕ is in axiom group 4. Consequently, ! ∀ x ϕ, as is evidenced by the tree:

Case 3: ϕ is obtained by modus ponens from ψ and ψ → ϕ. Then by inductive hypothesis we have ! ∀ x ψ and ! ∀ x(ψ → ϕ). This is just the situation in which axiom group 3 is useful. That ! ∀ x ϕ is evidenced by the tree:

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So by induction ! ∀ x ϕ for every theorem ϕ of .

(The sole reasons for having axiom groups 3 and 4 are indicated by the above proof.) The restriction that x not occur free in is essential. For example, P x |= ∀ x P x, and so by the soundness theorem to appear in Section 2.5, P x ∀ x P x. On the other hand, x will in general occur free in the formula ϕ. For example, at the beginning of this subsection we showed ﬁrst that ! (P x → ∃ y P y). The second example there, ! ∀ x(P x → ∃ y P y), was obtained from the ﬁrst example as in case 3 of the above proof. EXAMPLE.

∀ x ∀ y α ! ∀ y ∀ x α.

The proof of the generalization theorem actually yields somewhat more than was stated. It shows how we can, given a deduction of ϕ from , effectively transform it to obtain a deduction of ∀ x ϕ from . LEMMA 24C (RULE T) If ! α1 , . . . , ! αn and {α1 , . . . , αn } tautologically implies β, then ! β. PROOF. α1 → · · · → αn → β is a tautology, and hence a logical axiom. Apply modus ponens n times. DEDUCTION THEOREM If ; γ ! ϕ, then ! (γ → ϕ). (The converse clearly holds also; in fact, the converse is essentially the rule modus ponens.) FIRST PROOF ; γ ! ϕ

iff (; γ ) ∪ tautologically implies ϕ, iff ∪ tautologically implies (γ → ϕ), iff ! (γ → ϕ).

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SECOND PROOF The second proof does not use the compactness theorem of sentential logic as does the ﬁrst proof. It shows in a direct way how to transform a deduction of ϕ from ; γ to obtain a deduction of (γ → ϕ) from . We show by induction that for every theorem ϕ of ; γ the formula (γ → ϕ) is a theorem of . Case 1: ϕ = γ . Then obviously ! (γ → ϕ). Case 2: ϕ is a logical axiom or a member of . Then ! ϕ. And ϕ tautologically implies (γ → ϕ), whence by rule T we have ! (γ → ϕ). Case 3: ϕ is obtained by modus ponens from ψ and ψ → ϕ. By the inductive hypothesis, ! (γ → ψ) and ! (γ → (ψ → ϕ)). And the set {γ →ψ, γ →(ψ →ϕ)} tautologically implies γ →ϕ. Thus, by rule T, ! (γ → ϕ). So by induction the conclusion holds for any ϕ deducible from ; γ . COROLLARY 24D (CONTRAPOSITION)

; ϕ ! ¬ ψ iff ; ψ ! ¬ ϕ.

PROOF ; ϕ ! ¬ ψ ⇒ ! ϕ → ¬ ψ ⇒ ! ψ → ¬ϕ ⇒ ; ψ ! ¬ ϕ

by the deduction theorem, by rule T, by modus ponens.

(In the second step we use the fact that ϕ → ¬ ψ tautologically implies ψ → ¬ ϕ.) By symmetry, the converse holds also. Say that a set of formulas is inconsistent iff for some β, both β and ¬ β are theorems of the set. (In this event, any formula α is a theorem of the set, since β → ¬ β → α is a tautology.) COROLLARY 24E (REDUCTIO AD ABSURDUM) then ! ¬ ϕ.

If ; ϕ is inconsistent,

PROOF. From the deduction theorem we have ! (ϕ → β) and ! (ϕ → ¬ β). And {ϕ → β, ϕ → ¬ β} tautologically implies ¬ ϕ. EXAMPLE. ! ∃ x ∀ y ϕ → ∀ y ∃ x ϕ. There are strategic advantages to working backward. It sufﬁces to show that ∃ x ∀ y ϕ ! ∀ y ∃ x ϕ, by the deduction theorem.

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A Mathematical Introduction to Logic It sufﬁces to show that ∃ x ∀ y ϕ ! ∃ x ϕ, by the generalization theorem. It sufﬁces to show that ¬ ∀ x ¬ ∀ y ϕ ! ¬ ∀ x ¬ ϕ, as this is the same as the preceding. It sufﬁces to show that ∀ x ¬ ϕ ! ∀ x ¬ ∀ y ϕ, by contraposition (and rule T). It sufﬁces to show that ∀ x ¬ ϕ ! ¬ ∀ y ϕ, by generalization. It sufﬁces to show that {∀ x ¬ ϕ, ∀ y ϕ} is inconsistent, by reductio ad absurdum. And this is easy: 1. ∀ x ¬ ϕ ! ¬ ϕ by axiom group 2 and modus ponens. 2. ∀ y ϕ ! ϕ for the same reason. Lines 1 and 2 show that {∀ x ¬ ϕ, ∀ y ϕ} is inconsistent.

Strategy As the preceding example indicates, the generalization and deduction theorems (and to a smaller extent the corollaries) will be very useful in showing that certain formulas are deducible. But there is still the matter of strategy: For a given and ϕ, where should one begin in order to show that ! ϕ? One could, in principle, start enumerating all ﬁnite sequences of wffs until one encountered a deduction of ϕ from . Although this would be an effective procedure (for reasonable languages) for locating a deduction if one exists, it is far too inefﬁcient to have more than theoretical interest. One technique is to abandon formality and to give in English a proof that the truth of implies the truth of ϕ. Then the proof in English can be formalized into a legal deduction. (In the coming pages we will see techniques for carrying out such a formalization in a reasonably natural way.) There are also useful methods based solely on the syntactical form of ϕ. Assume then that ϕ is indeed deducible from but that you are seeking a proof of this fact. There are several cases: 1. Suppose that ϕ is (ψ → θ ). Then it will sufﬁce to show that ; ψ ! θ (and this will always be possible). 2. Suppose that ϕ is ∀ x ψ. If x does not occur free in , then it will sufﬁce to show that ! ψ. (Even if x should occur free in , the difﬁculty can be circumvented. There will always be a variable y such that ! ∀ y ψ yx and ∀ y ψ yx ! ∀ x ψ. See the re-replacement lemma, Exercise 9.) 3. Finally, suppose that ϕ is the negation of another formula. 3a. If ϕ is ¬(ψ → θ), then it will sufﬁce to show that ! ψ and ! ¬ θ (by rule T). And this will always be possible. 3b. If ϕ is ¬ ¬ ψ, then of course it will sufﬁce to show that ! ψ.

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3c. The remaining case is where ϕ is ¬ ∀ x ψ. It would sufﬁce to show that ! ¬ ψtx , where t is some term substitutable for x in ψ. (Why?) Unfortunately this is not always possible. There are cases in which ! ¬ ∀ x ψ, and yet for every term t, ¬ ψtx . (One such example is = ∅, ψ = ¬ (P x → ∀ y P y).) Contraposition is handy here; ; α ! ¬ ∀ x ψ iff ; ∀ x ψ ! ¬ α. (A variation on this is: ; ∀ y α ! ¬ ∀ x ψ if ; ∀ x ψ ! ¬ α.) If all else fails, one can try reductio ad absurdum. EXAMPLE (Q2A).

If x does not occur free in α, then ! (α → ∀ x β) ↔ ∀ x(α → β).

To prove this, it sufﬁces (by rule T) to show, that ! (α → ∀ x β) → ∀ x(α → β) and ! ∀ x(α → β) → (α → ∀ x β). For the ﬁrst of these, it sufﬁces (by the deduction and generalization theorems) to show that {(α → ∀ x β), α} ! β. But this is easy; ∀ x β → β is an axiom. To obtain the converse, ! ∀ x(α → β) → (α → ∀ x β), it sufﬁces (by the deduction and generalization theorems) to show that {∀ x(α → β), α} ! β. This again is easy. In the above example we can replace α by ¬ α, β by ¬ β, and use the contraposition tautology (among other things) to obtain the related fact:

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A Mathematical Introduction to Logic If x does not occur free in α, then

(Q3B).

! ( ∃ x β → α) ↔ ∀ x(β → α). The reader might want to convince him- or herself that the above formula is valid. Frequently an abbreviated style is useful in writing down a proof of deducibility, as in the following example. EXAMPLE (EQ2).

∀ x ∀ y(x = y → y = x).

PROOF ! x = y → x = x → y = x. Ax 6. ! x = x. Ax 5. ! x = y → y = x. 1, 2; T. ! ∀ x ∀ y(x = y → y = x). 3; gen2 .

1. 2. 3. 4.

In line 1, “Ax 6” means that the formula belongs to axiom group 6. In line 3, “1, 2; T” means that this line is obtained from lines 1 and 2 by rule T. In line 4, “3; gen2 ” means that the generalization theorem can be applied twice to line 3 to yield line 4. In the same spirit we write “MP,” “ded,” and “RAA” to refer to modus ponens, the deduction theorem, and reductio ad absurdum, respectively. It must be emphasized that the four numbered lines above do not constitute a deduction of ∀ x ∀ y(x = y → y = x). Instead they form a proof (in the meta-language we continue, with little justiﬁcation, to call English) that such a deduction exists. The shortest deduction of ∀ x ∀ y(x = y → y = x) known to the author is a sequence of 17 formulas. EXAMPLE.

! x = y → ∀ z P x z → ∀ z P yz.

PROOF 1. 2. 3. 4. 5. 6.

! x = y → P x z → P yz. Ax 6. ! ∀ z P x z → P x z. Ax 2. ! x = y → ∀ z P x z → P yz. 1, 2; T. {x = y, ∀ z P x z} ! P yz. 3; MP2 . {x = y, ∀ z P x z} ! ∀ z P yz. 4; gen. ! x = y → ∀ z P x z → ∀ z P yz. 5; ded2 .

EXAMPLE (EQ5).

Let f be a two-place function symbol. Then

! ∀ x1 ∀ x2 ∀ y1 ∀ y2 (x1 = y1 → x2 = y2 → f x1 x2 = f y1 y2 ). PROOF.

Two members of axiom group 6 are x1 = y1 → f x1 x2 = f x1 x2 → f x1 x2 = f y1 x2 , x2 = y2 → f x1 x2 = f y1 x2 → f x1 x2 = f y1 y2 .

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From ∀ x x = x (in axiom group 5) we deduce f x1 x2 = f x1 x2 . The three displayed formulas tautologically imply x1 = y1 → x2 = y2 → f x1 x2 = f y1 y2 .

EXAMPLE (a) {∀ x(P x → Qx), ∀ z Pz} ! Qc. It is not hard to show that such a deduction exists. The deduction itself consists of seven formulas. (b) {∀ x(P x → Qx), ∀ z Pz} ! Qy. This is just like (a). The point we are interested in here is that we can use the same sevenelement deduction, with c replaced throughout by y. (c) {∀ x(P x → Qx), ∀ z Pz} ! ∀ y Qy. This follows from (b) by the generalization theorem. (d) {∀ x(P x → Qx), ∀ z Pz} ! ∀ x Qx. This follows from (c) by use of the fact that ∀ y Qy ! ∀ x Qx. Parts (a) and (b) of the foregoing example illustrate a sort of interchangeability of constant symbols with free variables. This interchangeability is the basis for the following variation on the generalization theorem, for which part (c) is an example. Part (d) is covered by Corollary 24G. ϕ yc is, of course, the result of replacing c by y in ϕ. THEOREM 24F (GENERALIZATION ON CONSTANTS) Assume that ! ϕ and that c is a constant symbol that does not occur in . Then there is a variable y (which does not occur in ϕ) such that ! ∀ y ϕ yc . Furthermore, there is a deduction of ∀ y ϕ yc from in which c does not occur. PROOF. Let α0 , . . . , αn be a deduction of ϕ from . (Thus αn = ϕ.) Let y be the ﬁrst variable that does not occur in any of the αi ’s. We claim that (α0 )cy , . . . , (αn )cy

(∗)

is a deduction from of ϕ yc . So we must check that each (αk )cy is in ∪ or is obtained from earlier formulas by modus ponens. Case 1: αk ∈ . Then c does not occur in αk . So (αk )cy = αk , which is in . Case 2: αk is a logical axiom. Then (αk )cy is also a logical axiom. (Read the list of logical axioms and note that introducing a new variable will transform a logical axiom into another one.) Case 3: αk is obtained by modus ponens from αi and α j (which is (αi → αk )) for i, j less than k. Then (α j )cy = ((αi )cy → (αk )cy ). So (αk )cy is obtained by modus ponens from (αi )cy and (α j )cy .

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A Mathematical Introduction to Logic This completes the proof that (∗) above is a deduction of ϕ yc . Let be the ﬁnite subset of that is used in (∗). Thus (∗) is a deduction of ϕ yc from , and y does not occur in . So by the generalization theorem, ! ∀ y ϕ yc . Furthermore, there is a deduction of ∀ y ϕ yc from in which c does not appear. (For the proof to the generalization theorem did not add any new symbols to a deduction.) This is also a deduction from of ∀ y ϕ yc . We will sometimes want to apply this theorem in circumstances in which not just any variable will do. In the following version, there is a variable x selected in advance. COROLLARY 24G Assume that ! ϕcx , where the constant symbol c does not occur in or in ϕ. Then ! ∀ x ϕ, and there is a deduction of ∀ x ϕ from in which c does not occur. PROOF. By the above theorem we have a deduction (without c) from of ∀ y((ϕcx )cy ), where y does not occur in ϕcx . But since c does not occur in ϕ, x c ϕc y = ϕ yx . It remains to show that ∀ y ϕ yx ! ∀ x ϕ. We can easily do this if we know that (∀ y ϕ yx ) → ϕ is an axiom. That is, x must be substitutable for y in ϕ yx , and (ϕ yx )xy must be ϕ. This is reasonably clear; for details see the re-replacement lemma (Exercise 9). COROLLARY 24H (RULE EI) Assume that the constant symbol c does not occur in ϕ, ψ, or , and that ; ϕcx ! ψ. Then ; ∃ x ϕ ! ψ and there is a deduction of ψ from ; ∃ x ϕ in which c does not occur. PROOF.

By contraposition we have ; ¬ ψ ! ¬ ϕcx .

So by the preceding corollary we obtain ; ¬ ψ ! ∀ x ¬ ϕ. Applying contraposition again, we have the desired result.

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“EI” stands for “existential instantiation,” a bit of traditional terminology. We will not have occasion to use rule EI in any of our proofs, but it may be handy in exercises. It is the formal counterpart to the reasoning: “We know there is an x such that x . So call it c. Now from c we can prove ψ.” But notice that rule EI does not claim that ∃ x ϕ ! ϕcx , which is in fact usually false. EXAMPLE, revisited. ! ∃ x ∀ y ϕ → ∀ y ∃ x ϕ. By the deduction theorem, it sufﬁces to show that ∃ x ∀ y ϕ ! ∀ y ∃ x ϕ. By rule EI it sufﬁces to show that ∀ y ϕcx ! ∀ y ∃ x ϕ, where c is new to the language. By the generalization theorem it sufﬁces to show that ∀ y ϕcx ! ∃ x ϕ. Since ∀ y ϕcx ! ϕcx , it sufﬁces to show that ϕcx ! ∃ x ϕ. By contraposition this is equivalent to ∀ x ¬ ϕ ! ¬ ϕcx , which is trivial (by axiom group 2 and modus ponens). We can now see roughly how our particular list of logical axioms was formed. The tautologies were included to handle the sentential connective symbols. (We could economize considerably at this point by using only some of the tautologies.) Axiom group 2 reﬂects the intended meaning of the quantiﬁer symbol. Then in order to be able to prove the generalization theorem we added axiom groups 3 and 4 and arranged for generalizations of axioms to be axioms. Axiom groups 5 and 6 will turn out to be just enough to prove the crucial properties of equality; see the subsection on equality. As we will prove in Section 2.5, every logical axiom is a valid formula. It might seem simpler to use as logical axioms the set of all valid formulas. But there are two (related) objections to doing this. For one, the concept of validity was deﬁned semantically. That is, the deﬁnition referred to possible meanings (i.e., structures) for the language and to the concept of truth in a structure. For our present purposes (e.g., proving that the validities are effectively enumerable) we need a class with a ﬁnitary, syntactical deﬁnition. That is, the deﬁnition of involves only matters concerning the arrangement of the symbols in the logical axioms; there is no reference to matters of truth in structures. A second

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A Mathematical Introduction to Logic objection to the inclusion of all valid formulas as axioms is that we prefer a decidable set , and the set of validities fails to be decidable.

Alphabetic Variants Often when we are discussing a formula such as ∀ x(x = 0 → ∃ y x = Sy) we are not interested in the particular choice of the variables x and y. We want x, y to be a pair of distinct variables, but often it makes no difference whether the pair is v4 , v9 or v8 , v1 . But when it comes time to substitute a term t into a formula, then the choice of quantiﬁed variables can make the difference between the substitutability of t and its failure. In this subsection we will discuss what to do when substitutability fails. As will be seen, the difﬁculty can always be surmounted by suitably juggling the quantiﬁed variables. For example, suppose we want to show that ! ∀ x ∀ y P x y → ∀ y P yy. There is the difﬁculty that y is not substitutable for x in ∀ y P x y, so the above sentence is not in axiom group 2. This is a nuisance resulting from an unfortunate choice of variables. For example, showing that ! ∀ x ∀ z P x z → ∀ y P yy involves no such difﬁculties. So we can solve our original problem if we know that ! ∀ x ∀ y P x y → ∀ x ∀ z P x z, which, again, involves no difﬁculties. This slightly circuitous strategy (of interpolating ∀ x ∀ z P x z between ∀ x ∀ y P x y and ∀ y P yy) is typical of a certain class of problems. Say that we desire to substitute a term t for x in a wff ϕ. If t is, in fact, not so substitutable, then we replace ∀ x ϕ by ∀ x ϕ , where t is substitutable for x in ϕ . In the above example ϕ is ∀ y P x y and ϕ is ∀ z P x z. In general ϕ will differ from ϕ only in the choice of quantiﬁed variables. But ϕ must be formed in a reasonable way so as to be logically equivalent to ϕ. For example, it would be unreasonable to replace ∀ y P x y by ∀ x P x x, or ∀ y ∀ z Qx yz by ∀ z ∀ z Qx zz. THEOREM 24I (EXISTENCE OF ALPHABETIC VARIANTS) Let ϕ be a formula, t a term, and x a variable. Then we can ﬁnd a formula ϕ (which differs from ϕ only in the choice of quantiﬁed variables) such that (a) ϕ ! ϕ and ϕ ! ϕ; (b) t is substitutable for x in ϕ .

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PROOF. We consider ﬁxed t and x, and construct ϕ by recursion on ϕ. The ﬁrst cases are simple: For atomic ϕ we take ϕ = ϕ, and then (¬ ϕ) = (¬ ϕ ), (ϕ → ψ) = (ϕ → ψ ). But now consider the choice of ( ∀ y ϕ) . If y does not occur in t, or if y = x, then we can just take (∀ y ϕ) = ∀ y ϕ . But for the general case we must change the variable. Choose a variable z that does not occur in ϕ or t or x. Then deﬁne (∀ y ϕ) = ∀ z(ϕ )zy . To verify that (b) holds, we note that z does not occur in t and t is substitutable for x in ϕ (by the inductive hypothesis). Hence (since x =

z) t is also substitutable for x in (ϕ )zy . To verify that (a) holds, we calculate: ϕ ! ϕ

by the inductive hypothesis;

∴∀ y ϕ ! ∀ y ϕ . ∀ y ϕ ! (ϕ )zy ∴ ∀ y ϕ ! ∀ z(ϕ )zy ∴ ∀ y ϕ ! ∀ z(ϕ )zy .

since z does not occur in ϕ ; by generalization;

In the other direction, ∀ z(ϕ )zy ! ((ϕ )zy )zy , which is ϕ by Exercise 9; ϕ ! ϕ; by the inductive hypothesis; y ∴ ∀ z(ϕ )z ! ϕ ∴ ∀ z(ϕ )zy ! ∀ y ϕ

by generalization.

The last step uses the fact that y does not occur free in (ϕ )zy unless y = z, and so does not occur free in ∀ z(ϕ )zy in any case. The formulas ϕ constructed as in the proof of this theorem will be called alphabetic variants of ϕ. The moral of the theorem is: One should not be daunted by failure of substitutability; the right alphabetic variant will avoid the difﬁculty.

Equality We list here (assuming that our language includes =) the facts about equality that will be needed in the next section. First, the relation deﬁned by v1 = v2 is reﬂexive, symmetric, and transitive (i.e., is an equivalence relation): Eq1: PROOF.

! ∀ x x = x.

Axiom group 5. Eq2:

! ∀ x ∀ y(x = y → y = x).

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A Mathematical Introduction to Logic PROOF.

Page 122. ! ∀ x ∀ y ∀ z(x = y → y = z → x = z).

Eq3: PROOF.

Exercise 11.

In addition, we will need to know that equality is compatible with the predicate and function symbols: Eq4

(for a two-place predicate symbol P):

! ∀ x1 ∀ x2 ∀ y1 ∀ y2 (x1 = y1 → x2 = y2 → P x1 x2 → P y1 y2 ). Similarly for n-place predicate symbols. PROOF.

It sufﬁces to show that {x1 = y1 , x2 = y2 , P x1 x2 } ! P y1 y2 .

This is obtained by application of modus ponens to the two members of axiom group 6: x1 = y1 → P x1 x2 → P y1 x2 , x2 = y2 → P y1 x2 → P y1 y2 .

Eq5 (for a two-place function symbol f ): ! ∀ x1 ∀ x2 ∀ y1 ∀ y2 (x1 = y1 → x2 = y2 → f x1 x2 = f y1 y2 ). Similarly for n-place function symbols. PROOF.

Page 122.

Final Comments A logic book in the bootstrap tradition might well begin with this section on a deductive calculus. Such a book would ﬁrst state the logical axioms and the rules of inference and would explain that they are acceptable to reasonable people. Then it would proceed to show that many formulas were deducible (or deducible from certain nonlogical axioms, such as axioms for set theory). Our viewpoint is very different. We study, among other things, the facts about the procedure described in the preceding paragraph. And we employ in this any correct mathematical reasoning, whether or not such reasoning is known to have counterparts in the deductive calculus under study. Figure 8 is intended to illustrate the separation between (a) the level at which we carry out our reasoning and prove our results, and (b) the level of the deductive calculus which we study.

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Figure 8. The meta-language above, in which we study the object language below.

Exercises 1. For a term u, let u tx be the expression obtained from u by replacing the variable x by the term t. Restate this deﬁnition without using any form of the word “replace” or its synonyms. Suggestion: Use recursion on u. (Observe that from the new deﬁnition it is clear that u tx is itself a term.) 2. To which axiom groups, if any, do each of the following formulas belong? (a) [( ∀ x P x → ∀ y P y) → P z] → [∀ x P x → ( ∀ y P y → P z)]. (b) ∀ y[∀ x(P x → P x) → (Pc → Pc)]. (c) ∀ x ∃ y P x y → ∃ y P yy. 3. (a) Let A be a structure and let s : V → |A|. Deﬁne a truth assignment v on the set of prime formulas by v(α) = T

iff

|=A α[s].

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A Mathematical Introduction to Logic Show that for any formula (prime or not), v(α) = T

iff

|=A α[s].

Remark: This result reﬂects the fact that ¬ and → were treated in Chapter 2 the same way as in Chapter 1. (b) Conclude that if tautologically implies ϕ, then logically implies ϕ. 4. Give a deduction (from ∅) of ∀ x ϕ → ∃ x ϕ. (Note that you should not merely prove that such a deduction exists. You are instead asked to write out the entire deduction.) 5. Find a function f such that if a formula ϕ has a deduction of length n from a set , and if x does not occur free in , then ∀ x ϕ has a deduction from of length f (n). The more slowly your function grows, the better. Show that if ! α → β, then ! ∀ x α → ∀ x β. Show that it is not in general true that α → β |= ∀ x α → ∀ x β. Show that ! ∃ x(P x → ∀ x P x). Show that {Qx, ∀ y(Qy → ∀ z Pz)} ! ∀ x P x. 8. (Q2b) Assume that x does not occur free in α. Show that 6. (a) (b) 7. (a) (b)

! (α → ∃ x β) ↔ ∃ x(α → β). Also show that, under the same assumption, we have Q3a: ! ( ∀ x β → α) ↔ ∃ x(β → α). 9. (Re-replacement lemma) (a) Show by example that (ϕ yx )xy is not in general equal to ϕ. And that it is possible both for x to occur in (ϕ yx )xy at a place where it does not occur in ϕ, and for x to occur in ϕ at a place where it does not occur in (ϕ yx )xy . (b) Show that if y does not occur at all in ϕ, then x is substitutable for y in ϕ yx and (ϕ yx )xy = ϕ. Suggestion: Use induction on ϕ. 10. Show that ∀ x ∀ y P x y ! ∀ y ∀ x P yx. 11. (Eq3) Show that ! ∀ x ∀ y ∀ z(x = y → y = z → x = z). 12. Show that any consistent set of formulas can be extended to a consistent set having the property that for any formula α, either α ∈ or (¬ α) ∈ . (Assume that the language is countable. Do not use the compactness theorem of sentential logic.) 13. Show that ! P y ↔ ∀ x(x = y → P x). Remarks: More generally, if t is substitutable for x in ϕ and x does not occur in t, then ! [ϕtx ↔ ∀ x(x = t → ϕ)].

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14. 15.

16.

17.

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Thus the formula ∀ x(x = t → ϕ) offers an alternative of sorts to the substitution ϕtx . Show that ! ( ∀ x((¬ P x) → Qx) → ∀ y((¬ Qy) → P y)). Show that deductions (from ∅) of the following formulas exist: (a) ∃ x α ∨ ∃ x β ↔ ∃ x(α ∨ β). (b) ∀ x α ∨ ∀ x β → ∀ x(α ∨ β). Show that deductions (from ∅) of the following formulas exist: (a) ∃ x(α ∧ β) → ∃ x α ∧ ∃ x β. (b) ∀ x(α ∧ β) ↔ ∀ x α ∧ ∀ x β. Show that deductions (from ∅) of the following formulas exist: (a) ∀ x(α → β) → ( ∃ x α → ∃ x β). (b) ∃ x(P y ∧ Qx) ↔ P y ∧ ∃ x Qx.

SECTION 2.5 Soundness and Completeness Theorems In this section we establish two major theorems: the soundness of our deductive calculus ( ! ϕ ⇒ |= ϕ) and its completeness ( |= ϕ ⇒ ! ϕ). We will then be able to draw a number of interesting conclusions (including the compactness and enumerability theorems). Although our deductive calculus was chosen in a somewhat arbitrary way, the signiﬁcant fact is that some such deductive calculus is sound and complete. This should be encouraging to the “working mathematician” concerned about the existence of proofs from axioms; see the Retrospectus subsection of Section 2.6. SOUNDNESS THEOREM If ! ϕ, then |= ϕ. The soundness theorem tells that our deductions lead only to “correct” conclusions — deductions would be rather pointless otherwise! The idea of the proof is that the logical axioms are logically implied by anything, and that modus ponens preserves logical implications. LEMMA 25A Every logical axiom is valid. PROOF OF THE SOUNDNESS THEOREM, ASSUMING THE LEMMA. We show by induction that any formula ϕ deducible from is logically implied by . Case 1: ϕ is a logical axiom. Then by the lemma |= ϕ, so a fortiori |= ϕ. Case 2: ϕ ∈ . Then clearly |= ϕ. Case 3: ϕ is obtained by modus ponens from ψ and ψ → ϕ, where (by the inductive hypothesis) |= ψ and |= (ψ → ϕ). It then follows at once that |= ϕ.

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A Mathematical Introduction to Logic It remains, of course, to prove that lemma. We know from Exercise 6 of Section 2.2 that any generalization of a valid formula is valid. So it sufﬁces to consider only logical axioms that are not themselves generalizations of other axioms. We will examine the various axiom groups in order of complexity. Axiom group 3: See Exercise 3 of Section 2.2. Axiom group 4: See Exercise 4 of Section 2.2. Axiom group 5: Trivial. A satisﬁes x = x with s iff s(x) = s(x), which is always true. Axiom group 1: We know from Exercise 3 of the preceding section that if ∅ tautologically implies α, then ∅ |= α. And that is just what we need. Axiom group 6 (for an example, see Exercise 5 of Section 2.2): Assume that α is atomic and α is obtained from α by replacing x at some places by y. It sufﬁces to show that {x = y, α} |= α . So take any A, s such that |=A x = y[s],

i.e., s(x) = s(y).

Then any term t has the property that if t is obtained from t by replacing x at some places by y, then s(t) = s(t ). This is obvious; a full proof would use induction on t. If α is t1 = t2 , then α must be t1 = t2 , where ti is obtained from ti as described. |=A α[s] iff s(t1 ) = s(t2 ), iff s(t1 ) = s(t2 ), iff |=A α [s]. Similarly, if α is Pt1 · · · tn , then α is Pt1 · · · tn and an analogous argument applies. Finally, we come to axiom group 2. It will be helpful to consider ﬁrst a simple case: we will show that ∀ x P x → Pt is valid. Assume that |=A ∀ x P x[s]. Then for any d in |A|, |=A P x[s(x | d)]. So in particular we may take d = s(t): |=A P x[s(x | s(t))].

(a)

This is equivalent (by the deﬁnition of satisfaction of atomic formulas) to s(t) ∈ P A ,

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which in turn is equivalent to |=A Pt[s].

(b)

For this argument to be applicable to the nonatomic case, we need a way of passing from (a) to (b). This will be provided by the substitution lemma below, which states that |=A ϕ[s(x | s(t))]

iff |=A ϕtx [s]

whenever t is substitutable for x in ϕ. Consider a ﬁxed A and s. For any term u, let u tx be the result of replacing the variable x in u by the term t. LEMMA 25B s(u tx ) = s(x | s(t))(u). This looks more complicated than it is. It asserts that a substitution can be carried out either in the term u or in s, with equivalent results. The corresponding commutative diagram is shown.

PROOF. By induction on the term u. If u is a constant symbol or a variable other than x, then u tx = u and the desired equation reduces to s(u) = s(u). If u = x, then the equation reduces to s(t) = s(t). The inductive step, although cumbersome to write, is mathematically trivial. The substitution lemma is similar in spirit; it states that a substitution can be carried out either within ϕ or in s, with equivalent results. For an example see Exercise 10 of Section 2.2. SUBSTITUTION LEMMA If the term t is substitutable for the variable x in the wff ϕ, then |=A ϕtx [s]

iff |=A ϕ[s(x | s(t))].

PROOF. We use induction on ϕ to show that the above holds for every s. Case 1: ϕ is atomic. Then the conclusion follows from the preceding lemma. For example, if ϕ is Pu for some term u, then |=A Pu tx [s] iff s(u tx ) ∈ P A , iff s(x | s(t))(u) ∈ P A by Lemma 25B, iff |=A Pu[s(x | s(t))].

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A Mathematical Introduction to Logic Case 2: ϕ is ¬ ψ or ψ → θ . Then the conclusion for ϕ follows at once from the inductive hypotheses for ψ and θ . Case 3: ϕ is ∀ y ψ, and x does not occur free in ϕ. Then s and s(x | s(t)) agree on all variables that occur free in ϕ. And also ϕtx is simply ϕ. So the conclusion is immediate. Case 4: ϕ is ∀ y ψ, and x does occur free in ϕ. Because t is substitutable for x in ϕ, we know that y does not occur in t and t is substitutable for x in ψ (see the deﬁnition of “substitutable”). By the ﬁrst of these, s(t) = s(y | d)(t)

(∗)

for any d in |A|. Since x =

y, ϕtx = ∀ y ψtx . |=A ϕtx [s]

iff for every d, |=A ψtx [s(y | d)], iff for every d, |=A ψ[s(y | d)(x | s(t))] by the inductive hypothesis and (∗), iff |=A ϕ[s(x | s(t))].

So by induction the lemma holds for all ϕ.

Axiom group 2: Assume that t is substitutable for x in ϕ. Assume that A satisﬁes ∀ x ϕ with s. We need to show that |=A ϕtx [s]. We know that for any d in |A|, |=A ϕ[s(x | d)], In particular, let d = s(t): |=A ϕ[s(x | s(t))], So, by the substitution lemma, |=A ϕtx [s]. Hence ∀ x ϕ → ϕtx is valid. This completes the proof that all logical axioms are valid. And so the soundness theorem is proved. COROLLARY 25C lent.

If ! (ϕ ↔ ψ), then ϕ and ψ are logically equiva-

COROLLARY 25D If ϕ is an alphabetic variant of ϕ (see Theorem 24I), then ϕ and ϕ are logically equivalent. Recall that a set is consistent iff there is no formula ϕ such that both ! ϕ and ! ¬ ϕ. Deﬁne to be satisﬁable iff there is some A and s such that A satisﬁes every member of with s. COROLLARY 25E

If is satisﬁable, then is consistent.

This corollary is actually equivalent to the soundness theorem, as the reader is invited to verify.

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The completeness theorem is the converse to the soundness theorem and is a deeper result. ¨ , 1930) COMPLETENESS THEOREM (GODEL (a) If |= ϕ, then ! ϕ. (b) Any consistent set of formulas is satisﬁable. Actually parts (a) and (b) are equivalent; cf. Exercise 2. So it sufﬁces to prove part (b). We will give a proof for a countable language; later we will indicate what alterations are needed for languages of larger cardinality. (A countable language is one with countably many symbols, or equivalently, by Theorem 0B, one with countably many wffs.) The ideas of the proof are related to those in the proof of the compactness theorem for sentential logic. We begin with a consistent set . In steps 1–3 we extend to a set of formulas for which (i) ⊆ . (ii) is consistent and is maximal in the sense that for any formula α, either α ∈ or (¬ α) ∈ . (iii) For any formula ϕ and variable x, there is a constant c such that (¬ ∀ x ϕ → ¬ ϕcx ) ∈ . Then in step 4 we form a structure A in which members of not containing = can be satisﬁed. |A| is the set of terms, and for a predicate symbol P, t1 , . . . , tn ∈ P A

iff

Pt1 · · · tn ∈ .

Finally, in steps 5 and 6 we change A to accommodate formulas containing the equality symbol. It is suggested that on a ﬁrst reading the details which are provided for most of the steps be omitted. Once the outline is clearly in mind, the entire proof should be read. (The nondetails are marked with a stripe in the left margin.) PROOF.

Let be a consistent set of wffs in a countable language.

STEP 1: Expand the language by adding a countably inﬁnite set of new constant symbols. Then remains consistent as a set of wffs in the new language. Details: If not then for some β, there is a deduction (in the expanded language) of (β ∧ ¬ β) from . This deduction contains only ﬁnitely many of the new constant symbols. By the theorem for generalization on constants (Theorem 24F), each can be replaced by a variable. We then have a deduction (in the original language) of (β ∧ ¬ β ) from . This contradicts our assumption that was consistent.

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A Mathematical Introduction to Logic STEP 2: For each wff ϕ (in the new language) and each variable x, we want to add to the wff ¬ ∀ x ϕ → ¬ ϕcx , where c is one of the new constant symbols. (The idea is that c volunteers to name a counterexample to ϕ, if there is any.) We can do this in such a way that together with the set of all the added wffs is still a consistent set. Details: Adopt a ﬁxed enumeration of the pairs ϕ, x , where ϕ is a wff (of the expanded language) and x is a variable: ϕ1 , x1 , ϕ2 , x2 , ϕ3 , x3 , . . . . This is possible since the language is countable. Let θ1 be ¬ ∀ x1 ϕ1 → ¬ ϕ1 cx11 , where c1 is the ﬁrst of the new constant symbols not occurring in ϕ1 . Then go on to ϕ2 , x2 and deﬁne θ2 . In general, θn is ¬ ∀ xn ϕn → ¬ ϕn

xn cn ,

where cn is the ﬁrst of the new constant symbols not occurring in ϕn or in θk for any k < n. Let be the set {θ1 , θ2 , . . .}. We claim that ∪ is consistent. If not, then (because deductions are ﬁnite) for some m ≥ 0, ∪ {θ1 , . . . , θm , θm+1 } is inconsistent. Take the least such m. Then by RAA ∪ {θ1 , . . . , θm } ! ¬ θm+1 . Now θm+1 is ¬ ∀ x ϕ → ¬ ϕcx for some x, ϕ, and c. So by rule T, we obtain the two facts: ∪ {θ1 , . . . θm } ! ¬ ∀ x ϕ, ∪ {θ1 , . . . θm } ! ϕcx .

(∗)

Since c does not appear in any formula on the left side, we can apply the Corollary 24G to the second of these, obtaining ∪ {θ1 , . . . , θm } ! ∀ x ϕ. This and (∗) contradict the leastness of m (or the consistency of , if m = 0).

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STEP 3: We now extend the consistent set ∪ to a consistent set which is maximal in the sense that for any wff ϕ either ϕ ∈ or (¬ ϕ) ∈ . Details: We can imitate the proof used at the analogous place in the proof of sentential compactness in Section 1.7. Or we can argue as follows: Let be the set of logical axioms for the expanded language. Since ∪ is consistent, there is no formula β such that ∪ ∪ tautologically implies both β and ¬ β. (This is by Theorem 24B; the compactness theorem of sentential logic is used here.) Hence there is a truth assignment v for the set of all prime formulas that satisﬁes ∪ ∪ . Let = {ϕ | v(ϕ) = T }. Clearly for any ϕ either ϕ ∈ or ( ¬ ϕ) ∈ but not both. Also we have ! ϕ ⇒ tautologically implies ϕ ⇒ v(ϕ) = T ⇒ ϕ ∈ .

(since ⊆ ), since v satisﬁes ,

Consequently, is consistent, lest both ϕ and ( ¬ ϕ) belong to . Actually, regardless of how is constructed, it must be deductively closed. That is, ! ϕ ⇒ ¬ ϕ by consistency, ⇒ (¬ ϕ) ∈ / , ⇒ϕ∈ by maximality. STEP 4: We now make from a structure A for the new language, but with the equality symbol (if any) replaced by a new two-place predicate symbol E. A will not itself be the structure in which will be satisﬁed but will be a preliminary structure. (a) |A| = the set of all terms of the new language. (b) Deﬁne the binary relation E A by u, t ∈ E A

iff

the formula u = t belongs to .

(c) For each n-place predicate parameter P, deﬁne the n-ary relation P A by t1 , . . . , tn ∈ P A

iff

Pt1 · · · tn ∈ .

(d) For each n-place function symbol f , let f A be the function deﬁned by f A (t1 , . . . , tn ) = f t1 · · · tn .

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A Mathematical Introduction to Logic This includes the n = 0 case; for a constant symbol c we take cA = c. Deﬁne also a function s : V → |A|, namely the identity function s(x) = x on V . It then follows that for any term t, s(t) = t. For any wff ϕ, let ϕ ∗ be the result of replacing the equality symbol in ϕ by E. Then |=A ϕ ∗ [s]

iff

ϕ ∈ .

Details: That s(t) = t can be proved by induction on t, but the proof is entirely straightforward. The other claim, that |=A ϕ ∗ [s]

iff

ϕ ∈ ,

we prove by induction on the number of places at which connective or quantiﬁer symbols appear in ϕ. Case 1: Atomic formulas. We deﬁned A in such a way as to make this case immediate. For example, if ϕ is Pt, then |=A Pt[s]

iff s(t) ∈ P A , iff t ∈ P A , iff Pt ∈ .

Similarly, |=A u Et[s]

iff s(u), s(t) ∈ E A , iff u, t ∈ E A , iff u = t ∈ .

Case 2: Negation. |=A (¬ ϕ)∗ [s]

Case 3: Conditional. |=A (ϕ → ψ)∗ [s] iff iff iff ⇒ ⇒ ⇒

if | =A ϕ ∗ [s], iff ϕ ∈ / by inductive hypothesis, iff (¬ ϕ) ∈ by properties of .

|=A ϕ ∗ [s] or |=A ψ ∗ [s], ϕ∈ / or ψ ∈ by inductive hypothesis, (¬ ϕ) ∈ or ψ ∈ , ! (ϕ → ψ), in fact tautologically, ϕ∈ / or [ϕ ∈ and ! ψ], (¬ ϕ) ∈ or ψ ∈ ,

which closes the loop. And ! (ϕ → ψ)

iff (ϕ → ψ) ∈ .

(This should be compared with Exercise 2 of Section 1.7.) Case 4: Quantiﬁcation. We want to show that |=A ∀ x ϕ ∗ [s]

iff ∀ x ϕ ∈ .

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(The notational ambiguity is harmless since ∀ x(ϕ ∗ ) is the same as (∀ x ϕ)∗ .) includes the wff θ: ¬ ∀ x ϕ → ¬ ϕcx . To show that |=A ∀ x ϕ ∗ [s] ⇒ ∀ x ϕ ∈ , we can argue: If ϕ ∗ is true of everything, then it is true of c, whence by the inductive hypothesis ϕcx ∈ . But then ∀ x ϕ ∈ , because c was chosen to be a counterexample to ϕ if there was one. In more detail: |=A ∀ x ϕ ∗ [s] ⇒ |=A ϕ ∗ [s(x | c)] ⇒ |=A (ϕ ∗ )cx [s] ⇒ |=A (ϕcx )∗ [s], ⇒ ϕcx ∈ / ⇒ (¬ ϕcx ) ∈ ⇒ (¬ ∀ x ϕ) ∈ /

by the substitution lemma this being the same formula by the inductive hypothesis by consistency since θ ∈ and is deductively closed

⇒ ∀ x ϕ ∈ . (This is our only use of . We needed to know that if (¬ ∀ x ϕ) ∈ , then for a particular c we would have (¬ ϕcx ) ∈ .) We turn now to the converse. We can almost argue as follows:

|=A ∀ x ϕ ∗ [s] ⇒ |=A ϕ ∗ [s(x | t)] for some t by the substitution lemma |=A (ϕtx )∗ [s] / by the inductive hypothesis ⇒ ϕtx ∈ ∀x ϕ ∈ / since is deductively closed. The ﬂaw here is that the two wavy implications require that t be substitutable for x in ϕ. This may not be the case, but we can use the usual repair: We change to an alphabetic variant ψ of ϕ in which t is substitutable for x. Then

|=A ∀ x ϕ ∗ [s] ⇒ |=A ϕ ∗ [s(x | t)] for some t, henceforth ﬁxed ⇒ |=A ψ ∗ [s(x | t)] by the semantical equivalence of alphabetic variants (Corollary 25D) by the substitution lemma ⇒ |=A (ψtx )∗ [s] / by the inductive hypothesis ⇒ ψtx ∈ ⇒ ∀x ψ ∈ / since is deductively closed ⇒ ∀x ϕ ∈ / by the syntactical equivalence of alphabetic variants (Theorem 24I). This completes the list of possible cases; it now follows by induction that for any ϕ, |=A ϕ ∗ [s]

iff

ϕ ∈ .

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A Mathematical Introduction to Logic If our original language did not include the equality symbol, then we are done. For we need only restrict A to the original language to obtain a structure that satisﬁes every member of with the identity function. But now assume that the equality symbol is in the language. Then A will no longer serve. For example, if contains the sentence c = d (where c and d are distinct constant symbols), then we need a structure B in which cB = d B . We obtain B as the quotient structure A/E of A modulo E A . STEP 5: E A is an equivalence relation on |A|. For each t in |A| let [t] be its equivalence class. E A is, in fact, a congruence relation for A. This means that the following conditions are met: (i) E A is an equivalence relation on |A|. (ii) P A is compatible with E A for each predicate symbol P: t1 , . . . , tn ∈ P A and ti E A ti for 1 ≤ i ≤ n ⇒ t1 , . . . , tn ∈ P A . (iii) f A is compatible with E A for each function symbol f : ti E A ti for 1 ≤ i ≤ n ⇒ f A (t1 , . . . , tn )E A f A (t1 , . . . , tn ). Under these circumstances we can form the quotient structure A/E, deﬁned as follows: (a) |A/E| is the set of all equivalence classes of members of |A|. (b) For each n-place predicate symbol P, [t1 ], . . . , [tn ] ∈ P A/E iff t1 , . . . , tn ∈ P A . (c) For each n-place function symbol f , f A/E ([t1 ], . . . , [tn ]) = [ f A (t1 , . . . , tn )]. This includes the n = 0 cases: cA/E = [cA ]. Let h : |A| → |A/E| be the natural map: h(t) = [t]. Then h is a homomorphism of A onto A/E. Furthermore, E A/E is the equality relation on |A/E|. Consequently, for any ϕ: ϕ ∈ ⇔ |=A ϕ ∗ [s] ⇔ |=A/E ϕ ∗ [h ◦ s] ⇔ |=A/E ϕ[h ◦ s] So A/E satisﬁes every member of (and hence every member of ) with h ◦ s.

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Details: Recall that t E At

iff (t = t ) ∈ , iff ! t = t .

(i) E A is an equivalence relation on A by properties Eq1, Eq2, and Eq3 of equality. (ii) P A is compatible with E A by property Eq4 of equality. (iii) f A is compatible with E A by property Eq5 of equality. It then follows from the compatibility of P A with E A that P A/E is well deﬁned. Similarly, f A/E is well deﬁned because f A is compatible with E A . It is immediate from the construction that h is a homomorphism of A onto A/E. And [t]E A/E [t ]

iff t E A t , iff [t] = [t ].

Finally, ϕ ∈ ⇔ |=A ϕ ∗ [s] by step 4 ⇔ |=A/E ϕ ∗ [h ◦ s] by the homomorphism theorem ⇔ |=A/E ϕ[h ◦ s], the last step being justiﬁed by the fact that E A/E is the equality relation on |A/E|. STEP 6: Restrict the structure A/E to the original language. This restriction of A/E satisﬁes every member of with h ◦ s. For an uncountable language, a few modiﬁcations to the foregoing proof of the completeness theorem are needed. Say that the language has cardinality λ. (By this we mean that it has λ symbols or, equivalently, λ formulas.) We will describe the modiﬁcations needed, assuming the reader has a substantial knowledge of set theory. In step 1 we add λ new constant symbols; the details remain unchanged. In step 2, only the details change. The cardinal λ is an initial ordinal. (We have tacitly well ordered the language here.) “Enumerate” the pairs ϕα , xα α<λ indexed by ordinals less than λ. For α < λ, θα is ¬ ∀ xα ϕα → (¬ ϕ)cxαα , where cα is the ﬁrst of the new constant symbols not in ϕα or in θβ for any β < α. (This excludes at most ℵ0 · card(α) constant symbols, so there are some left.) Finally, in step 3, we can obtain the maximal set by use of Zorn’s lemma. The rest of the proof remains unchanged.

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A Mathematical Introduction to Logic COMPACTNESS THEOREM (a) If |= ϕ, then for some ﬁnite 0 ⊆ we have 0 |= ϕ. (b) If every ﬁnite subset 0 of is satisﬁable, then is satisﬁable. In particular, a set of sentences has a model iff every ﬁnite subset has a model. PROOF To prove part (a) of the compactness theorem, we simply observe that |= ϕ ⇒ ! ϕ ⇒ 0 ! ϕ for some ﬁnite 0 ⊆ , deductions being ﬁnite ⇒ 0 |= ϕ. Part (b) has a similar proof. If every ﬁnite subset of is satisﬁable, then by soundness every ﬁnite subset of is consistent. Thus is consistent, since deductions are ﬁnite. So by completeness, is satisﬁable. (Actually parts (a) and (b) are equivalent; cf. Exercise 3 of Section 1.7.) When a person ﬁrst hears of the compactness theorem, his natural inclination is to try to combine (by some algebraic or set-theoretic operation) the structures in which the various ﬁnite subsets are satisﬁed, in such a way as to obtain a structure in which the entire set is satisﬁed. In fact, such a proof is possible; the operation to use is the ultraproduct construction. But we will refrain from digressing further into this intriguing possibility. Notice that the compactness theorem involves only semantical notions of Section 2.2; it does not involve deductions at all. And there are proofs that avoid deductions. The same remarks apply to the following theorem.

ENUMERABILITY THEOREM For a reasonable language, the set of valid wffs can be effectively enumerated.

By a reasonable language we mean one whose set of parameters can be effectively enumerated and such that the two relations { P, n | P is an n-place predicate symbol} and { f, n | f is an n-place function symbol} are decidable. For example, any language with only ﬁnitely many parameters (such a language will be called a ﬁnite language) is certainly reasonable, because ﬁnite sets are always decidable. On the other hand, a reasonable language must be countable, since we cannot effectively enumerate an uncountable set. (In fact, a stronger statement applies: As

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in Section 1.7, a suitable input/output format is needed in which the underlying set of symbols communicated is ﬁnite, which implies that the set of all strings is countable.) A precise version of this theorem will be given in Section 3.4. (See especially item 20 there.) The proofs of the two versions are in essence the same. PROOF. The essential fact is that , and hence the set of deductions, is decidable. Suppose that we are given some expression ε. (The assumption of reasonableness enters already here. There are only countably many things eligible to be given by one person to another.) We want to decide whether or not ε is in . First we check that ε has the syntactical form necessary to be a formula. (For sentential logic we gave detailed instructions for such a check; see Section 1.3. Similar instructions can be given for ﬁrst-order languages, by using Section 2.3.) If ε passes that test, we then check (by constructing a truth table) to see if ε is a generalization of a tautology. If not, then we proceed to see if ε has the syntactical form necessary to be in axiom group 2. And so forth. If ε has not been accepted by the time we ﬁnish with axiom group 6, then ε is not in . (The above is intended to convince the reader that he really can tell members of from nonmembers. The reader who remains dubious can look forward to the rerun in Section 3.4.) Since is decidable, the set of tautological consequences of is effectively enumerable; see Theorem 17G. But {α | α is a tautological consequence of } = {α | ! α} by Theorem 24B, = {α | α is valid}.

An alternative to the last paragraph of this proof is the following argument, which is possibly more illuminating: First we claim that the set of deductions (from ∅) is decidable. For given a ﬁnite sequence α0 , . . . , αn we can examine each αi in turn to see if it is in or is obtainable by modus ponens from earlier members of the sequence. Then to enumerate the validities, we begin by enumerating all ﬁnite sequences of wffs. We look at each sequence as it is produced and decide whether or not it is a deduction. If not, we discard it. But if it is, then we put its last member on the list of validities. Continuing in this way, we generate — in an inefﬁcient way — a list on which any valid formula will eventually appear.

COROLLARY 25F Let be a decidable set of formulas in a reasonable language.

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A Mathematical Introduction to Logic (a) The set of theorems of is effectively enumerable. (b) The set {ϕ | |= ϕ} of formulas logically implied by is effectively enumerable. (Of course parts (a) and (b) refer to the same set. This corollary includes the enumerability theorem itself, in which = ∅.) PROOF 1. form

Enumerate the validities; whenever you ﬁnd one of the αn → · · · → α 1 → α 0 ,

check to see if αn , . . . , α1 are in . If so, then put α0 on the list of theorems of . In this way, any theorem of is eventually listed. PROOF 2. ∪ is decidable, so its set of tautological consequences is effectively enumerable. And that is just the set we want. For example, let be the (decidable) set of axioms for any of the usual systems of set theory. Then this corollary tells us that the set of theorems of set theory is effectively enumerable. More generally, in setting up some axiomatic theory, it is natural to insist that the set of axioms be decidable. After all, we want proofs from these axioms to be convincing arguments that can be veriﬁed. Part of the veriﬁcation process involves checking that statements alleged to be axioms are indeed axioms. For this to be possible, the set of axioms needs to be decidable (or at least semidecidable). This has the consequence that the set of theorems that follow from the axioms is effectively enumerable.

COROLLARY 25G Assume that is a decidable set of formulas in a reasonable language, and for any sentence σ either |= σ or |= ¬ σ . Then the set of sentences implied by is decidable.

PROOF. If is inconsistent, then we have simply the (decidable) set of all sentences. So assume that is consistent. Suppose that we are given a sentence σ and asked to decide whether or not |= σ . We can enumerate the theorems of and look for σ or ¬ σ . Eventually one will appear, and then we know the answer. (Observe that this proof actually describes two decision procedures. One is correct when is inconsistent, the other is correct when is consistent. So in either case a decision procedure exists. But we cannot necessarily determine effectively, given a ﬁnite description of , which one is to be used. A set is decidable if there exists a decision procedure for it. That is not the same as having a known decision procedure in our hands.)

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It should be remarked that our proofs of enumerability cannot, in general, be strengthened to proofs of decidability. For almost all languages the set of validities is not decidable. (See Church’s Theorem, Section 3.5.)

Historical Notes The completeness theorem (for countable languages) was contained in the 1930 doctoral dissertation of Kurt G¨odel. (It is not to be confused with the “G¨odel incompleteness theorem,” published in 1931. We will consider this latter result in Chapter 3.) The compactness theorem (for countable languages) was given as a corollary. The compactness theorem for uncountable languages was implicit in a 1936 paper by Anatolii Mal cev. His proof used Skolem functions (cf. Section 4.2) and the compactness theorem of sentential logic. The ﬁrst explicit statement of the compactness theorem for uncountable languages was in a 1941 paper by Mal cev. The enumerability theorem, as well as following from G¨odel’s 1930 work, was also implicit in results published in 1928 by Thoralf Skolem. The proof we have given for the completeness theorem is patterned after one given by Leon Henkin in his dissertation, published in 1949. Unlike G¨odel’s original proof, Henkin’s proof generalizes easily to languages of any cardinality.

Exercises 1. (Semantical rule EI) Assume that the constant symbol c does not occur in ϕ, ψ, or , and that ; ϕcx |= ψ. Show (without using the soundness and completeness theorems) that ; ∃ x ϕ |= ψ. 2. Prove the equivalence of parts (a) and (b) of the completeness theorem. Suggestion: |= ϕ iff ∪ {¬ ϕ} is unsatisﬁable. And is satisﬁable iff |= ⊥, where ⊥ is some unsatisﬁable, refutable formula like ¬ ∀ x x = x. Remark: Similarly, the soundness theorem is equivalent to the statement that every satisﬁable set of formulas is consistent. 3. Assume that ! ϕ and that P is a predicate symbol which occurs neither in nor in ϕ. Is there a deduction of ϕ from in which P nowhere occurs? Suggestion: There are two very different approaches to this problem. The “soft” approach makes use of two languages, one that contains P and one that does not. The “hard” approach considers the question whether P can be systematically eliminated from a given deduction of ϕ from .

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A Mathematical Introduction to Logic 4. Let = {¬ ∀ v1 Pv1 , Pv2 , Pv3 , . . .}. Is consistent? Is satisﬁable? 5. Show that an inﬁnite map can be colored with four colors iff every ﬁnite submap of it can be. Suggestion: Take a language having a constant symbol for each country and having four one-place predicate symbols for the colors. Use the compactness theorem. 6. Let 1 and 2 be sets of sentences such that nothing is a model of both 1 and 2 . Show that there is a sentence τ such that Mod 1 ⊆ Mod τ and Mod 2 ⊆ Mod ¬ τ . (This can be stated: Disjoint EC classes can be separated by an EC class.) Suggestion: 1 ∪ 2 is unsatisﬁable; apply compactness. 7. The completeness theorem tells us that each sentence either has a deduction (from ∅) or has a counter-model (i.e., a structure in which it is false). For each of the following sentences, either show there is a deduction or give a counter-model. (a) ∀ x(Qx → ∀ y Qy) (b) ( ∃ x P x → ∀ y Qy) → ∀ z(P z → Qz) (c) ∀ z(P z → Qz) → ( ∃ x P x → ∀ y Qy) (d) ¬ ∃ y ∀ x(P x y ↔ ¬ P x x) 8. Assume the language (with equality) has just the parameters ∀ and P, where P is a two-place predicate symbol. Let A be the structure with |A| = Z, the set of integers (positive, negative, and zero) and with a, b ∈ P A iff |a − b| = 1. Thus A looks like an inﬁnite graph: · · · ←→ • ←→ • ←→ • ←→ · · · Show that there is an elementarily equivalent structure B that is not connected. (Being connected means that for every two members of |B|, there is a path between them. A path — of length n — from a to b is a sequence p0 , p1 , . . . , pn with a = p0 and b = pn and pi , pi+1 ∈ P B for each i.) Suggestion: Add constant symbols c and d. Write down sentences saying c and d are far apart. Apply compactness. 9. In Section 2.4 we used a certain set of logical axioms. That set can be altered, within limits. (a) Suppose we add to some formula ψ that is not valid. Show that the soundness theorem now fails. (b) At the other extreme, suppose we take no logical axioms at all: = ∅. Show that the completeness theorem now fails. (c) Suppose we modify by adding one new valid formula. Explain why both the soundness theorem and the completeness theorem still hold.

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SECTION 2.6 Models of Theories In this section we will leave behind deductions and logical axioms. Instead we return to topics discussed in Section 2.2. But now, in the presence of the theorems of the preceding section, we will be able to answer more questions than we could before.

Finite Models Some sentences have only inﬁnite models, for example, the sentence saying that < is an ordering with no largest element. The negation of such a sentence is ﬁnitely valid, that is, it is true in every ﬁnite structure. It is also possible to have sentences having only ﬁnite models. For example, any model of ∀ x ∀ y x = y has cardinality 1. But if all models of are ﬁnite, then there is a ﬁnite bound on the size of the models, by the following theorem. THEOREM 26A If a set of sentences has arbitrarily large ﬁnite models, then it has an inﬁnite model. PROOF. For each integer k ≥ 2, we can ﬁnd a sentence λk that translates, “There are at least k things.” For example, v2 , λ2 = ∃ v1 ∃ v2 v1 = v2 ∧ v1 = v3 ∧ v2 = v3 ). λ3 = ∃ v1 ∃ v2 ∃ v3 (v1 = Consider the set ∪ {λ2 , λ3 , . . .}. By hypothesis any ﬁnite subset has a model. So by compactness the entire set has a model, which clearly must be inﬁnite. For example, it is a priori conceivable that there might be some very subtle equation of group theory that was true in every ﬁnite group but false in every inﬁnite group. But by the above theorem, no such equation exists. The proof to this theorem illustrates a useful method for obtaining a structure with given properties. One writes down sentences (possibly in an expanded language) stating the properties one wants. One then argues that any ﬁnite subset of the sentences has a model. The compactness theorem does the rest. We will see more examples of this method in the coming pages. COROLLARY 26B The class of all ﬁnite structures (for a ﬁxed language) is not EC . The class of all inﬁnite structures is not EC.

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A Mathematical Introduction to Logic PROOF. The ﬁrst sentence follows immediately from the theorem. If the class of all inﬁnite structures is Mod τ , then the class of all ﬁnite structures is Mod ¬ τ . But this class isn’t even EC , much less EC. The class of inﬁnite structures is EC , being Mod{λ2 , λ3 , . . .}. Next we want to consider decision problems connected with ﬁnite structures. For any structure A, deﬁne the theory of A, written Th A, to be the set of all sentences true in A. For a ﬁnite structure A, is Th A decidable? Is the set of sentences having ﬁnite models decidable? The following observations will help here: 1. Any ﬁnite structure A is isomorphic to a structure with universe {1, 2, . . . , n} where n is the size of A (i.e., n = card |A|). The idea here is, where |A| = {a1 , . . . , an }, simply to replace ai by i. For example, suppose the language has only the parameters ∀ and a two-place predicate symbol E (for the “edge” relation in a directed graph). Consider the ﬁnite structure B with universe |B| consisting of a set of four distinct objects {a, b, c, d}, and with E B = { a, b , b, a , b, c , c, c }. Then B is isomorphic to the structure ({1, 2, 3, 4}; { 1, 2 , 2, 1 , 2, 3 , 3, 3 }). But there are other possibilities here; if we had focused on the members of |B| in the order b, a, d, c then we would have arrived at the isomorphic (but different) structure ({1, 2, 3, 4}; { 1, 2 , 2, 1 , 1, 4 , 4, 4 }). 2. A ﬁnite structure of the sort just described can, for a ﬁnite language, be speciﬁed by a ﬁnite string of symbols. In the example, ({1, 2, 3, 4}; { 1, 2 , 2, 1 , 2, 3 , 3, 3 }), the above line completely speciﬁes the structure, and it can be written down with numerals in base 10 (or your favorite base) along with punctuation and delimiters (e.g., parentheses). Therefore such a structure can be communicated to another person or to a machine. The ﬁnite string of symbols can be written down in a suitable input format. 3. Given a ﬁnite structure A for a ﬁnite language, with universe {1, . . . , n} (and by the preceding observation, it is possible to be given such an object), a wff ϕ, and an assignment sϕ of numbers in this universe to the variables free in ϕ (there are only ﬁnitely many, of course), we can effectively decide whether or not |=A ϕ[sϕ ].

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For example, given B = ({1, 2, 3, 4}; { 1, 2 , 2, 1 , 2, 3 , 3, 3 }) ϕ = ∀ v1 (( ¬ ∀ v2 ¬ Ev2 v1 ) → Ev1 v1 ), we can organize the computation in the tree form shown in Fig. 9. We ﬁnd that the sentence ϕ, which says “Anything in the range of E is related to itself,” is false in B.

Figure 9. Checking the sentence ∀v1 ((¬∀v2 ¬Ev2 v1 ) → Ev1 v1 ) in a structure with universe of size 4.

At each leaf in the tree (i.e., each minimal vertex), we have an atomic formula and we do a “table look-up” to see if it is satisﬁed. Notice that each quantiﬁer triggers a search through the n-element universe. For a formula ϕ with k quantiﬁers, the number of leaves in the tree will be bounded by a polynomial in n of degree k. If the language contains function symbols, then each term needs to be evaluated, using the (ﬁnite) function provided by the structure. In particular, restricting ourselves to sentences, we can effectively decide, given A as above and a sentence σ , whether or not A is a model of σ . (In fact here σ could even be a second-order sentence, as in Chapter 4.)

THEOREM 26C decidable.

For a ﬁnite structure A in a ﬁnite language, Th A is

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A Mathematical Introduction to Logic PROOF 1. By observation 1, we can replace A by an isomorphic structure with universe of the form {1, . . . , n}, without changing which sentences are true. Then apply observation 3. PROOF 2. By Exercise 17(a) of Section 2.2, there is a sentence δA that speciﬁes A up to isomorphism. It follows that Th A = {σ | δA |= σ }. (Details: For “⊆” note that if σ is true in A, then it is true in all isomorphic copies, and hence in all models of δA . So δA |= σ . The other direction is simpler; if δA |= σ then σ is true in all models of δA , of which A is one.) Apply Corollary 25G, noting that for each σ , either |=A σ or |=A ¬ σ . 4. Given a sentence σ and a positive integer n, we can effectively decide whether or not σ has an n-element model. That is, the binary relation { σ, n | σ has a model of size n} is decidable. The key idea is that there are only ﬁnitely many structures to check, and we can do that. The sentence σ has a model of size n if and only if it has a model with universe {1, . . . , n} by observation 1. With the language restricted to the parameters that occur in σ , there are only ﬁnitely many such structures, and we can systematically generate all of them. (For example, if the only parameters are ∀ and a 2-place predicate 2 symbol, then there are 2n different structures.) Applying observation 3, we test to see if any of these are models of σ . 5. The spectrum of a sentence σ is deﬁned to be {n | σ has a model of size n}. See Exercise 16 in Section 2.2. It follows from observation 4 that the spectrum of any sentence is a decidable set of positive integers.

THEOREM 26D For a ﬁnite language, {σ | σ has a ﬁnite model} is effectively enumerable.

PROOF. Here is a semidecision procedure: Given σ , ﬁrst check to see if it has a model of size 1, by using observation 4. If not, try size 2. Keep going.

COROLLARY 26E Assume the language is ﬁnite, and let be the set of sentences true in every ﬁnite structure. Then its complement, , is effectively enumerable.

PROOF.

For a sentence σ , σ ∈ ⇐⇒ (¬ σ ) has a ﬁnite model.

We can apply the above semidecision procedure to (¬ σ ).

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It follows (by Theorem 17F) that is decidable if and only if it is effectively enumerable. But this does not happen. We state without proof the following:

TRAKHTENBROT’S THEOREM (1950)

The set of sentences

= {σ | σ is true in every ﬁnite structure} is not in general decidable or effectively enumerable. Thus the analogue of the enumerability theorem for ﬁnite structures only is false.

Size of Models In the proof of the completeness theorem in Section 2.5, we started with a consistent set and formed a structure A/E in which it was satisﬁed. How large was that structure? We claim that if our initial language was countable, then |A/E| is a countable set. Thus a consistent set of sentences in a countable language has a countable model. A/E was constructed from a preliminary structure A. The universe of A was the set of all terms in the language obtained by adding a countable set of new constant symbols. But the augmented language was still countable, so the set of all expressions (and hence the set of all terms) was countable. That is, |A| was countable. The universe of A/E consisted of equivalence classes of members of A, so it, too, is a countable set. (We can map |A/E| one-to-one into |A| by assigning to each equivalence class some chosen member.) The conclusion is that A/E is a countable structure, as claimed. –SKOLEM THEOREM (1915) (a) Let be a satisﬁable set LOWENHEIM ¨ of formulas in a countable language. Then is satisﬁable in some countable structure. (b) Let be a set of sentences in a countable language. If has any model, then it has a countable model. PROOF. First observe that is consistent, by the soundness theorem. Then by the completeness theorem (plus the foregoing remarks) it can be satisﬁed in a countable structure. (There is another, more direct proof of this theorem that will be indicated in Section 4.2; see especially Exercise 1 there. That proof, which does not use a deductive calculus, begins with an arbitrary structure A in which can be satisﬁed, and by various manipulations extracts from it a suitable countable substructure in which is still satisﬁed.) The L¨owenheim–Skolem theorem was published by Leopold L¨owenheim in 1915 for the case where is a singleton; Thoralf Skolem in 1920 extended this to a possibly inﬁnite . The theorem marked a new phase in mathematical logic. Earlier work had been done in the direction

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A Mathematical Introduction to Logic of formalizing mathematics by means of formal languages and deductive calculi; this work was initiated largely by Gottlob Frege in 1879. For example, the Principia Mathematica (1910–1913) of Whitehead and Russell carried out such a formalization in great detail. But the modern phase began when logicians stepped back and began to prove results about the formal systems they had been constructing. Other early work in this trend was done by David Hilbert, Emil Post, Kurt G¨odel (as mentioned before), Alfred Tarski, and others. For a sample application of the L¨owenheim–Skolem theorem, let AST be your favorite set of axioms for set theory. We certainly hope these axioms are consistent. And so they have some model. By the L¨owenheim–Skolem theorem, the axioms have a countable model S. Of course, S is also a model of all the sentences logically implied by AST . One of these sentences asserts (when translated back into English according to the intended translation) that there are uncountably many sets. There is no contradiction here, but the situation is sufﬁciently puzzling to be called “Skolem’s paradox.” It is true that in the structure S there is no point that satisﬁes the formal deﬁnition of being a oneto-one map of the natural numbers onto the universe. But this in no way excludes the possibility of there being (outside S) some genuine function providing such a one-to-one correspondence. Recall that the theory of A, written Th A, is the set of all sentences true in A. We can apply the L¨owenheim–Skolem theorem (with = Th A) to prove that for any structure A for a countable language, there is a countable elementarily equivalent structure B. If B is a model of Th A then A ≡ B because |=A σ ⇒ σ ∈ Th A ⇒ |=B σ and

|=A σ ⇒ |=A ¬ σ ⇒ ( ¬ σ ) ∈ Th A ⇒ |=B ¬ σ ⇒ |=B σ. For example, the real ﬁeld (R; 0, 1, +, ·) is an uncountable structure for a ﬁnite language. Therefore there must be some countable structure (also a ﬁeld) satisfying exactly the same sentences. (In fact Tarski showed we can take the ﬁeld of algebraic real numbers for this.) EXAMPLE.

Consider the structure N = (N; 0, S, <, +, ·).

We claim that there is a countable structure M0 , elementarily equivalent to N (so that M0 and N satisfy exactly the same sentences) but not isomorphic to N. PROOF. We will construct M0 by using the compactness theorem. Expand the language by adding a new constant symbol c. Let = {0 < c, S0 < c, SS0 < c, . . .}.

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We claim that ∪ Th N has a model. For consider a ﬁnite subset. That ﬁnite subset is true in Nk = (N; 0, S, <, +, ·, k) (where k = cNk ) for some large k. So by the compactness theorem ∪ Th N has a model. By the L¨owenheim–Skolem theorem, ∪Th N has a countable model

M = |M|; 0M , SM ,

M0 = |M|; 0M , SM ,

|=N σ ⇒ ¬ σ ∈ Th N ⇒ |=M0 ¬ σ ⇒ |=M0 σ. We leave it to the reader to verify that M0 is not isomorphic to N. (|M0 | contains the “inﬁnite” number cM .) What about uncountable1 languages? Assume that in the proof of the completeness theorem we started with a set in a language of cardinality λ. We claim that in this case, the structure A/E we made has cardinality ≤λ. A/E was constructed from that preliminary structure A. The universe of A was the set of all terms in the language obtained by adding λ new constant symbols. So the augmented language still had cardinality λ. So (by Theorem 0D) the set of all expressions (and hence the set of all terms) had cardinality ≤λ. (In fact, because we had at least the λ new constant symbols, the set of terms had cardinality exactly λ.) The universe of A/E consisted of equivalence classes of members of A, so card |A/E| ≤ card |A|. (We can map |A/E| one-to-one into |A| by assigning to each equivalence class some chosen member, but now we may need the axiom of choice.) Thus, when the smoke had cleared, was satisﬁed in a structure A/E of cardinality ≤λ. LOWENHEIM ¨ –SKOLEM THEOREM (a) Let be a satisﬁable set of formulas in a language of cardinality λ. Then is satisﬁable in some structure of size ≤λ. (b) Let be a set of sentences in a language of cardinality λ. If has any model, then it has a model of cardinality ≤λ.

1

The reader who wishes to avoid uncountable cardinals is advised to skip to the subsection Theories.

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A Mathematical Introduction to Logic The earlier version of the L¨owenheim–Skolem theorem is a special case of this version, wherein λ = ℵ0 . Suppose that we have an uncountable structure A for a countable language. By the L¨owenheim–Skolem theorem (applied to Th A) there is a countable B that is a model of Th A, and hence A ≡ B, as noted previously. Conversely, suppose that we start with a countable structure B. Is there an uncountable A such that A ≡ B? If B is ﬁnite (and the language includes equality), then this is impossible. But if B is inﬁnite, then there will be such an A, by the following “upward and downward L¨owenheim– Skolem theorem.” The upward part is due to Tarski, whence the “T” of “LST.” LST THEOREM Let be a set of formulas in a language of cardinality λ, and assume that is satisﬁable in some inﬁnite structure. Then for every cardinal κ ≥ λ, there is a structure of cardinality κ in which is satisﬁable. PROOF. Let A be the inﬁnite structure in which is satisﬁable. Expand the language by adding a set C of κ new constant symbols. Let c2 | c1 , c2 distinct members of C}. = {c1 = Then every ﬁnite subset of ∪ is satisﬁable in the structure A, expanded to assign distinct objects to the ﬁnitely many new constant symbols in the subset. (Since A is inﬁnite, there is room to accommodate any ﬁnite number of these.) So by compactness ∪ is satisﬁable, and by the L¨owenheim–Skolem theorem it is satisﬁable in a structure B of cardinality ≤κ. (The expanded language has cardinality λ + κ = κ.) But any model of clearly has cardinality ≥κ. So B has cardinality κ; restrict B to the original language. COROLLARY 26F (a) Let be a set of sentences in a countable language. If has some inﬁnite model, then has models of every inﬁnite cardinality. (b) Let A be an inﬁnite structure for a countable language. Then for any inﬁnite cardinal λ, there is a structure B of cardinality λ such that B ≡ A. PROOF. (a) Take = , λ = ℵ0 in the theorem. (b) Take = Th A in part (a).

Consider a set of sentences, to be thought of as nonlogical axioms. (For example, might be a set of axioms for set theory or a set of axioms for number theory.) Call categorical iff any two models of are isomorphic. The above corollary implies that if has any

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inﬁnite models, then is not categorical. There is, for example, no set of sentences whose models are exactly the structures isomorphic to (N; 0, S, +, ·). This is indicative of a limitation in the expressiveness of ﬁrst-order languages. (As will be seen in Section 4.1, there are categorical second-order sentences. But second-order sentences are peculiar objects, obtained at the cost of holding the notion of subset ﬁxed, immune from interpretation by structures.)

Theories We deﬁne a theory to be a set of sentences closed under logical implication. That is, T is a theory iff T is a set of sentences such that for any sentence σ of the language, T |= σ ⇒ σ ∈ T. (Note that we admit only sentences, not formulas, with free variables.) For example, there is always a smallest theory, consisting of the valid sentences of the language. At the other extreme there is the theory consisting of all the sentences of the language; it is the only unsatisﬁable theory. For a class K of structures (for the language), deﬁne the theory of K (written Th K) by the equation Th K = {σ | σ is true in every member of K}. (This concept arose previously in the special case K = {A}.) THEOREM 26G

Th K is indeed a theory.

PROOF. Any member of K is a model of Th K. Thus if σ is true in every model of Th K, then it is true in every member of K. Whence it belongs to Th K. For example, if the parameters of the language are ∀, 0, 1, +, and ·, and F is the class of all ﬁelds, then Th F, the theory of ﬁelds, is simply the set of all sentences of the language which are true in all ﬁelds. If F0 is the class of ﬁelds of characteristic 0, then Th F0 is the theory of ﬁelds of characteristic 0. Recall that for a set of sentences, we deﬁned Mod to be the class of all models of . Th Mod is then the set of all sentences which are true in all models of . But this is just the set of all sentences logically implied by . Call this set the set of consequences of , Cn . Thus Cn = {σ | |= σ } = Th Mod . For example, set theory is the set of consequences of a certain set of sentences, known, unsurprisingly, as axioms for set theory. A set T of sentences is a theory iff T = Cn T .

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A Mathematical Introduction to Logic A theory T is said to be complete iff for every sentence σ , either σ ∈ T or (¬ σ ) ∈ T . For example, for any one structure A, Th{A} (written, as before, “Th A”) is always a complete theory. In fact, it is clear upon reﬂection that Th K is a complete theory iff any two members of K are elementarily equivalent. And a theory T is complete iff any two models of T are elementarily equivalent. For example, the theory of ﬁelds is not complete, since the sentences 1 + 1 = 0, ∃x x ·x =1+1 are true in some ﬁelds but false in others. The theory of algebraically closed ﬁelds of characteristic 0 is complete, but this is by no means obvious. (See Theorem 26J.) ∗

DEFINITION. A theory T is axiomatizable iff there is a decidable set of sentences such that T = Cn .

DEFINITION. A theory T is ﬁnitely axiomatizable iff T = Cn for some ﬁnite set of sentences. In the latter case we have T = Cn{σ } (written “T = Cn σ ”), where σ is the conjunction of the ﬁnitely many members of . For example, the theory of ﬁelds is ﬁnitely axiomatizable. For the class F of ﬁelds is Mod , where is the ﬁnite set of ﬁeld axioms. And the theory of ﬁelds is Th Mod = Cn . The theory of ﬁelds of characteristic 0 is axiomatizable, being Cn 0 , where 0 consists of the (ﬁnitely many) ﬁeld axioms together with the inﬁnitely many sentences: 1+1= 0, 1+1+1= 0, .... This theory is not ﬁnitely axiomatizable. To prove this, ﬁrst note that no ﬁnite subset of 0 has the entire theory as its set of consequences. (For that ﬁnite subset would be true in some ﬁeld of very large characteristic.) Then apply the following: THEOREM 26H If Cn is ﬁnitely axiomatizable, then there is a ﬁnite 0 ⊆ such that Cn 0 = Cn . PROOF. Say that Cn is ﬁnitely axiomatizable; then Cn = Cn τ for some one sentence τ . In general τ ∈ / , but at least |= τ . (τ ∈ Cn τ = Cn .) By the compactness theorem there is a ﬁnite 0 ⊆ such that 0 |= τ . Then Cn τ ⊆ Cn 0 ⊆ Cn , whence equality holds.

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We can now restate Corollaries 25F and 25G in the present terminology:

COROLLARY 26I (a) An axiomatizable theory (in a reasonable language) is effectively enumerable. (b) A complete axiomatizable theory (in a reasonable language) is decidable.

We can represent the relationships among these concepts by means of a diagram (in which we have included the results of Exercise 6):

For example, a theory that is given in axiomatic form (such as Zermelo–Fraenkel set theory, which is Cn AZF for a certain set AZF ) is effectively enumerable. We will argue in Section 3.7 that set theory (if consistent) is not decidable and not complete. Number theory, the theory of the structure (N; 0, S, <, +, ·, E), is complete but is not effectively enumerable and hence not axiomatizable (Section 3.5). We can use part (b) of the preceding corollary to establish the decidability of an axiomatizable theory, provided we can show that the theory in question is complete. This can sometimes be done by means of the Ło´s–Vaught test for completeness. Say that a theory T is ℵ0 -categorical iff all the inﬁnite countable models of T are isomorphic. More generally, for a cardinal κ, say that T is κ-categorical iff all models of T having cardinality κ are isomorphic. ŁOS´–VAUGHT TEST (1954) Let T be a theory in a countable language. Assume that T has no ﬁnite models. (a) If T is ℵ0 -categorical, then T is complete. (b) If T is κ-categorical for some inﬁnite cardinal κ, then T is complete. PROOF. It sufﬁces to show for any two models A and B of T that A ≡ B. Since A and B are inﬁnite, there exist (by the LST theorem) structures A ≡ A and B ≡ B having cardinality κ. A is isomorphic to B , so we have A ≡ A ∼ = B ≡ B. Thus A ≡ B.

(If T is a theory in a language of cardinality λ, then we must demand that λ ≤ κ.)

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A Mathematical Introduction to Logic The converse to the Ło´s–Vaught test is false. That is, there are complete theories which are not κ-categorical for any κ. In Section 3.1 we will apply the Ło´s–Vaught test to prove the decidability of the theory of the natural numbers with zero and successor. It can also be used to prove the decidability of the theory of the complex ﬁeld. But this proof will utilize an excursion into algebra. THEOREM 26J (a) The theory of algebraically closed ﬁelds of characteristic 0 is complete.

(b) The theory of the complex ﬁeld C = (C; 0, 1, +, ·) is decidable. PROOF. Let A be the class of algebraically closed ﬁelds of characteristic 0. Then A = Mod(0 ∪ ), where 0 consists as before of the axioms for ﬁelds of characteristic 0, and consists of the sentences ∀ a ∀ b ∀ c(a = 0 → ∃ x a · x · x + b · x + c = 0), ∀ a ∀ b ∀ c ∀ d(a = 0 → ∃ x a · x · x · x + b · x · x + c · x + d = 0), .... The set 0 ∪ is decidable and Th A = Cn(0 ∪ ), so this theory is axiomatizable. Part (a) of the theorem asserts that the theory is also complete, whence it is decidable. Part (b) follows from part (a). For we have C ∈ A, whence Th A ⊆ Th C. The completeness of Th A implies that equality holds; see Exercise 2. To prove part (a), we apply the Ło´s–Vaught test. The models of Th A are exactly the members of A. These are all inﬁnite. We further claim that Th A is categorical in any uncountable cardinality. This is equivalent to saying that any two algebraically closed ﬁelds of characteristic 0 having the same uncountable cardinality are isomorphic. This last assertion is a known result of algebra. We will sketch the proof, for the interest of those readers familiar with this topic. Any ﬁeld F is obtainable in the following way: (1) One begins with the prime subﬁeld, which is determined to within isomorphism by the characteristic of F. (2) One takes a transcendental extension, determined within isomorphism by the cardinality of the transcendence basis, i.e., by the transcendence degree of F (over its prime subﬁeld). (3) One ﬁnally takes some algebraic extension. We thus have a theorem of Steinitz: Two algebraically closed ﬁelds are isomorphic iff they have the same characteristic and the same transcendence degree. If the transcendence degree of an inﬁnite ﬁeld F is κ, then the cardinality of F is the larger of κ and ℵ0 . Hence for an uncountable

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ﬁeld, the cardinality equals the transcendence degree. So we conclude from Steinitz’s theorem that two algebraically closed ﬁelds having the same characteristic and the same uncountable cardinality are isomorphic. The theory of the real ﬁeld (R; 0, 1, +, ·) is also decidable. But this result (which is due to Tarski) is much deeper than the above theorem. The theory of the real ﬁeld is not categorical in any inﬁnite cardinality, so the Ło´s–Vaught test cannot be applied. As a ﬁnal application, we can show that the ordering of the rationals is elementarily equivalent to the ordering of the reals, (Q; < Q ) ≡ (R; < R ), where Q and R are the rationals and reals, respectively, and < Q and < R are the corresponding orderings. To show elementary equivalence, we show that both are models of some complete theory (which then must coincide with the theory of each structure). The key fact is provided by a theorem of Cantor: Any two countable dense linear orderings without endpoints are isomorphic. To give the details, we must back up a little. The language here has equality and the parameters ∀ and <. Let δ be the conjunction of the following sentences: 1. Ordering axioms (trichotomy and transitivity): ∀ x ∀ y(x < y ∨ x = y ∨ y < x), ∀ x ∀ y(x < y → y < x), ∀ x ∀ y ∀ z(x < y → y < z → x < z). 2. Density: ∀ x ∀ y(x < y → ∃ z(x < z < y)). 3. No endpoints: ∀ x ∃ y ∃ z(y < x < z). The dense linear orderings without endpoints are, by deﬁnition, the structures for this language that are models of δ. It is clear that they are all inﬁnite. Furthermore, we claim that the theory of these orderings, Cn δ, is ℵ0 -categorical. This is provided by the following fact. THEOREM 26K (CANTOR) (Q, < Q ).

Any countable model of δ is isomorphic to

We leave the proof to Exercise 4.

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A Mathematical Introduction to Logic We can now apply the Ło´s–Vaught test to conclude that Cn δ is complete. Hence any two models of δ are elementarily equivalent; in particular, (Q; < Q ) ≡ (R; < R ). We can also conclude that these structures have decidable theories.

Prenex Normal Form It will at times be convenient to move all the quantiﬁer symbols to the left of other symbols. For example, ∀ x(Ax → ∀ y Bx y) is equivalent to ∀ x ∀ y(Ax → Bx y). And ∀ x(Ax → ∃ y Bx y) is equivalent to ∀ x ∃ y(Ax → Bx y). Deﬁne a prenex formula to be one of the form (for some n ≥ 0) Q 1 x1 · · · Q n xn α, where Q i is ∀ or ∃ and α is quantiﬁer-free. PRENEX NORMAL FORM THEOREM For any formula, we can ﬁnd a logically equivalent prenex formula. PROOF. We will make use of the following quantiﬁer manipulation rules. Q1a. Q1b. Q2a. Q2b. Q3a. Q3b.

¬ ∀ x α |==| ∃ x ¬ α. ¬ ∃ x α |==| ∀ x ¬ α. (α → ∀ x β) |==| ∀ x(α → β) for x (α → ∃ x β) |==| ∃ x(α → β) for x ( ∀ x α → β) |==| ∃ x(α → β) for x ( ∃ x α → β) |==| ∀ x(α → β) for x

not free in α. not free in α. not free in β. not free in β.

Q1 is clear; for the others see the examples in Section 2.4 and Exercise 8 there. We now show by induction that every formula has an equivalent prenex formula. 1. For atomic formulas this is vacuous, as any quantiﬁer-free formula is trivially a prenex formula. 2. If α is equivalent to the prenex α , then ∀ x α is equivalent to the prenex ∀ x α .

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3. If α is equivalent to the prenex α , then ¬ α is equivalent to ¬ α . Apply Q1 to ¬ α to obtain a prenex formula; for example, ¬ ∀ x ∃ y ∃ z β |==| ∃ x ∀ y ∀ z ¬ β. 4. Finally we come to the case of α → β. By inductive hypothesis we have prenex formulas α and β equivalent to α and β, respectively. By our theorems on alphabetic variants, we may further assume that any variable that occurs quantiﬁed in one of the formulas α and β does not occur at all in the other. We then use Q2 and Q3 to obtain a prenex formula equivalent to α → β (and hence to α → β). Observe that there is some latitude in the order in which the rules Q2 and Q3 are applied. For example, ∀x ∃ y ϕ → ∃u ψ (where x and y do not occur in ψ, u does not occur in ϕ) is equivalent to any of the following: ∃ x ∀ y ∃ u(ϕ → ψ), ∃ x ∃ u ∀ y(ϕ → ψ), ∃ u ∃ x ∀ y(ϕ → ψ).

Retrospectus At the beginning of this book it was stated that symbolic logic is a mathematical model of deductive thought. This is as good a time as any to reﬂect on that statement, in the light of the material treated thus far. As a ﬁrst example, consider a mathematician working in set theory. He uses a language with an equality symbol, a symbol ∈ for membership, and numerous deﬁned symbols (∅, ∪, and others). In principle the deﬁned symbols could be eliminated and any sentence replaced by an equivalent sentence in which the deﬁned symbols did not appear. (In this connection see Section 2.7, where this topic is treated systematically.) He takes as primitive (or undeﬁned) notions the concepts of set and membership. He adopts some set AST of axioms involving these concepts. He asserts that for certain sentences (his theorems), these sentences are true provided the axioms are true, regardless of what the undeﬁned notions of set and membership actually mean. In support of these assertions he offers proofs, which are ﬁnitely long arguments intended to convince his colleagues of the correctness of the assertions. In terms of ﬁrst-order logic we can describe all this as follows: The language here is a ﬁrst-order language with equality and a two-place predicate symbol ∈. Thus ∀ and ∈ are the only parameters open to interpretation. There is a certain set AST of sentences in this language

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A Mathematical Introduction to Logic singled out as being the set of (nonlogical) axioms. Then certain other sentences are logical consequences of AST , i.e., are true in any model of AST . If τ is a consequence of AST (and only then), there is a deduction of τ from AST . Next consider a more typical case of the hypothetical working mathematician, that of the algebraist or analyst. The algebraist uses axioms for (say) group theory, but he also employs some amount of set theory. Similarly, the analyst deals with sentences that involve both numbers and sets of numbers. In both cases it is generally recognized that one could, in principle, convert the assertions of algebra and analysis to assertions of set theory. And then the remarks of the preceding paragraph again apply. The interest that symbolic logic holds for the mathematician is largely due to the accuracy with which it mirrors mathematical deductions. In the long run, it will surely be useful to understand the fundamental processes of doing mathematics. There remains the question of the accuracy with which ﬁrst-order logic mirrors nonmathematical deductive thought. Logic, symbolic and nonsymbolic, has always formed a traditional part of the philosophical study of the process by which people come to hold certain ideas. Nonmathematical examples to which ﬁrst-order logic applies are provided by a vast array of frivolous situations. Lewis Carroll gave such examples, one of which inferred that babies cannot manage crocodiles from the three hypotheses: (1) Babies are illogical. (2) Nobody is despised who can manage a crocodile. (3) Illogical persons are despised. But what of nonfrivolous situations? Here the applicability is obscured by the fact that we usually do not make explicit the assumptions we use in drawing conclusions. There are speciﬁc areas (in diverse ﬁelds such as physics, medicine, and law) where assumptions not only can be made explicit but are being made explicit. In some cases it appears that less than the full versatility of ﬁrst-order logic is required to formalize the real-life deductions. Possibly in other cases — ranging from everyday life to quantum mechanics — more features may be necessary.

Exercises 1. Show that the following sentences are ﬁnitely valid (i.e., that they are true in every ﬁnite structure): (a) ∃ x ∃ y ∃ z[(P x f x → P x x) ∨ (P x y ∧ P yz ∧ ¬ P x z)] (b) ∃ x ∀ y ∃ z[(Qzx → Qzy) → (Qx y → Qx x)] Suggestion: Show that any model of the negation must be inﬁnite. 2. Let T1 and T2 be theories (in the same language) such that (i) T1 ⊆ T2 , (ii) T1 is complete, and (iii) T2 is satisﬁable. Show that T1 = T2 .

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3. Establish the following facts: (a) 1 ⊆ 2 ⇒ Mod 2 ⊆ Mod 1 . K1 ⊆ K2 ⇒ Th K2 ⊆ Th K1 .

4.

5.

∗

6.

7.

8.

9.

10.

(b) ⊆ Th Mod and K ⊆ Mod Th K. (c) Mod = Mod Th Mod and Th K = Th Mod ThK. (Part (c) follows from (a) and (b).) Prove that any two countable dense linear orderings without endpoints are isomorphic (Theorem 26K). Suggestions: Let A and B be such structures with |A| = {a0 , a1 , . . .} and |B| = {b0 , b1 , . . .}. Construct an isomorphism in stages; at stage 2n be sure an is paired with some suitable b j , and at stage 2n + 1 be sure bn is paired with some suitable ai . Find prenex formulas equivalent to the following. (a) (∃ x Ax ∧ ∃ x Bx) → C x. (b) ∀ x Ax ↔ ∃ x Bx. Prove the converse to part (a) of Corollary 26I: An effectively enumerable theory (in a reasonable language) is axiomatizable. Suggestion: The set {σ0 , σ1 , σ2 , . . .} is equivalent (in the sense of having the same models) to the set {σ0 , σ0 ∧ σ1 , σ0 ∧ σ1 ∧ σ2 , . . .}. Consider a language with a two-place predicate symbol <, and let N = (N; <) be the structure consisting of the natural numbers with their usual ordering. Show that there is some A elementarily equivalent to N such that

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A Mathematical Introduction to Logic (b) Show that the set of valid ∀2 sentences is decidable. (A ∀2 formula is one of the form ∀ x1 · · · ∀ xm ∃ y1 · · · ∃ yn θ where θ is quantiﬁer-free.) Remarks: In ﬁrst-order logic, the “decision problem” (Entscheidungsproblem) is the problem of deciding, given a formula, whether or not it is valid. By Church’s theorem (Section 3.5), this problem in general is unsolvable. This exercise gives a solvable subcase of the decision problem.

SECTION 2.7 Interpretations Between Theories1 In some cases a theory T1 can be shown to be every bit as powerful as another theory T0 . This is certainly the case if the theories are in the same language and T0 ⊆ T1 . But even if the theories are in different languages, there may exist a way of translating from one language to the other in such a way that members of T0 are translated as members of T1 . This sort of situation will be examined in this section. We will begin by discussing the topic of deﬁned symbols. This topic, as well as having signiﬁcant interest of its own, will serve as an example for the situation of the preceding paragraph, wherein T0 is constructed from T1 by adding a new deﬁned symbol. If the deﬁnition is done properly, the original theory T1 should in principle be just as strong as the new T0 . We will consider only the case of deﬁned function symbols, since the case of deﬁned predicate symbols presents, in comparison, no real difﬁculties.

Deﬁning Functions Frequently in mathematics it is useful to introduce deﬁnitions of new functions. For example, in set theory one deﬁnes the power-set operation P by a sentence like, “Let P x be the set whose members are the subsets of x.” Or by a sentence in a formal language (here containing ∈, ⊆, and P), ∀ v1 ∀ v2 [Pv1 = v2 ↔ ∀ u(u ∈ v2 ↔ u ⊆ v1 )]. Now deﬁnitions are unlike theorems and unlike axioms. Unlike theorems, deﬁnitions are not things we prove. We just declare them by ﬁat. But unlike axioms, we do not expect deﬁnitions to add substantive information. A deﬁnition is expected to add to our convenience, not to our knowledge.

1

The results of this section will be used only in the latter part of Section 3.7.

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If this expectation is to be realized, the deﬁnition must be made in a reasonable way. As an example of a most unreasonable deﬁnition in number theory, suppose that we introduce a new function symbol by the “deﬁnition” f (x) = y

iff

x < y.

(Or by the sentence in a formal language: ∀ v1 ∀ v2 ( f v1 = v2 ↔ v1 < v2 ).) Since we know that 1 < 2, we see that f (1) = 2. But also 1 < 3, so we obtain f (1) = 3. And so we come to the conclusion (which does not itself involve f ) that 2 = 3. Obviously this deﬁnition of f was in some way very bad. It did not just make matters convenient for us; it enabled us to conclude that 2 = 3, which we could not have done without the deﬁnition. The trouble was that the deﬁnition bestowed the name “ f (1)” ambiguously upon many things (2 and 3 among them). Thus f (1) was not “well deﬁned.” Names ought to designate unique objects. In this subsection we want to consider conditions under which we can be assured that a deﬁnition will be satisfactory. To simplify the notation, we will consider only the deﬁnition of a one-place function symbol f , but the remarks will apply to n-place function symbols as well. Consider then a theory T in a language not yet containing the oneplace function symbol f . (For example, T might be the set of consequences of your favorite axioms for set theory.) We want to add f to the language, introducing it by the deﬁnition ∀ v1 ∀ v2 [ f v1 = v2 ↔ ϕ],

(δ)

where ϕ is a formula in the original language (i.e., a formula not containing f ) in which only v1 and v2 may occur free. THEOREM 27A

In the above situation, the following are equivalent:

(a) (The deﬁnition is noncreative.) For any sentence σ in the smaller language, if T ; δ |= σ (in the augmented language), then already T |= σ . (b) ( f is well deﬁned.) The sentence ∀ v1 ∃!v2 ϕ

(ε)

is in the theory T . (Here “∃!v2 ϕ” is an abbreviation for a longer formula; see Exercise 21 of Section 2.2.) PROOF. To obtain (a) ⇒ (b), simply note that δ |= ε. So by taking σ = ε in part (a), we obtain T |= ε. Conversely, assume that T |= ε. Let A be a model of T . (A is a structure for the original language.) For d ∈ |A|, let F(d)

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A Mathematical Introduction to Logic be the unique e ∈ |A| such that |=A ϕ[[d, e]]. (There is a unique such e because |=A ε.) Let (A, F) be the structure for the augmented language that agrees with A on the original parameters and that assigns F to the symbol f . Then it is easy to see that (A, F) is a model of δ. Furthermore, A and (A, F) satisfy the same sentences of the original language. In particular (A, F) is a model of T . Hence T ; δ |= σ ⇒ |=(A,F) σ ⇒ |=A σ.

(This argument can be stated more brieﬂy by using second-order logic. ε is logically equivalent to the sentence ∃ f δ.)

Interpretations The basic idea is that it is possible for one theory to be just as strong (in a sense to be made precise) as another theory in another language. In considering two languages simultaneously, there is no point in allowing them to conﬂict; e.g., the negation symbol of one language should not be a predicate symbol in the other. We can eliminate such conﬂicts by assuming that each of the languages is obtained from a third parent language by deleting some parameters (and perhaps equality). For example, axiomatic set theory is at least as strong as the theory of the natural numbers with zero and successor, i.e., the theory of (N; 0, S). Any sentence in the language of (N; 0, S) can be translated in a natural way into a sentence of set theory. (This translation is sketched brieﬂy in Section 3.7.) If the original sentence was true in (N; 0, S), then the translation will be a consequence of the axioms of set theory. (This is not obvious; the proof uses facts to be developed in Section 3.1.) Let us look more carefully at a second example. Consider on the one hand the theory of (N; 0, S) in its language, and on the other hand the theory of (Z; +, ·) in its language. (Here Z is the set of all integers, positive, negative, and zero.) We will shortly be in a position to claim that the second theory is as strong as the ﬁrst. How might a sentence about the natural numbers N with 0 and S be translated into a sentence about the integers Z with addition and multiplication? The ﬁrst clue is provided by Lagrange’s theorem from number theory: an integer is nonnegative iff it is the sum of four squares. Thus a quantiﬁer ∀ x in the ﬁrst language (where x is intended to range over N)

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can be replaced by ∀ x( ∃ y1 ∃ y2 ∃ y3 ∃ y4 x = y1 · y1 + y2 · y2 + y3 · y3 + y4 · y4 → in the second language. The second clue is that {0} and the successor function (viewed as a relation) are deﬁnable in (Z; +, ·). The set {0} is deﬁned by v1 + v1 = v1 . The successor relation (extended to Z) is deﬁned by ∀ z(z · z = z ∧ z + z = z → v1 + z = v2 ). Thus the sentence about (N; 0, S) ∀ x Sx = 0 can be translated into ∀ x[∃ y1 ∃ y2 ∃ y3 ∃ y4 x = y1 · y1 + y2 · y2 + y3 · y3 + y4 · y4 → ¬ ∀ u(u + u = u → ∀ v( ∀ z(z · z = z ∧ z + z = z→x+z = v) → v = u))]. So much for examples. For our general discussion it will be helpful to introduce the notation ϕ(t) = ϕtv1 , v ϕ(t1 , t2 ) = ϕtv1 1 t22 , and so forth. Thus ϕ = ϕ(v1 ) = ϕ(v1 , v2 ). If we use “ϕ(x)” we will not worry too much about whether or not x is substitutable for v1 in ϕ. If it is not, then we actually want ϕ(x) to be ψxv1 , where ψ is a suitable alphabetic variant of ϕ. Assume now that we have the following general situation: L 0 is a language. (A language can for all practical purposes be a set of parameters, possibly augmented by the equality symbol.) T1 is a theory in a (possibly different) language L 1 , which includes equality. DEFINITION. An interpretation π of L 0 into T1 is a function on the set of parameters of L 0 such that 1. π assigns to ∀ a formula π∀ of L 1 in which at most v1 occurs free, such that (i)

T1 |= ∃ v1 π∀ .

(The idea is that in any model of T1 , the formula π∀ should deﬁne a nonempty set to be used as the universe of an L 0 -structure.) 2. π assigns to each n-place predicate parameter P a formula π P of L 1 in which at most the variables v1 , . . . vn occur free.

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A Mathematical Introduction to Logic 3. π assigns to each n-place function symbol f a formula π f of L 1 , in which at most v1 , . . . , vn , vn+1 occur free, such that (ii)

T1 |= ∀ v1 · · · ∀ vn (π∀ (v1 ) → · · · → π∀ (vn )→ ∃ x(π∀ (x) ∧ ∀ vn+1 (π f (v1 , . . . , vn+1 ) ↔ vn+1 = x))).

(In English, this formula becomes, “For all v in the set deﬁned by π∀ , there is a unique x such that π f ( v , x), and furthermore x is in the set deﬁned by π∀ .” The idea is to ensure that in any model of T1 , π f deﬁnes a function on the universe deﬁned by π∀ . In the case of a constant symbol c, we have n = 0 and (ii) becomes T1 |= ∃ x(π∀ (x) ∧ ∀ v1 (πc (v1 ) ↔ v1 = x)). In other words, πc deﬁnes a singleton whose one member is also in the set deﬁned by π∀ .) For example, if L 0 is the language of (N; 0, S) and T1 is the theory of (Z; +, ·), then we have π∀ (x) = ∃ y1 ∃ y2 ∃ y3 ∃ y4 x = y1 · y1 + y2 · y2 + y3 · y3 + y4 · y4 , π0 (x) = x + x = x, z → x + z = y). πS (x, y) = ∀z (z · z = z ∧ z + z = (We are here exploiting the fact that in (Z; +, ·) we can, in effect, deﬁne the structure (N; 0, S).) If L 0 coincides with L 1 , there is trivially the identity interpretation π, for which π∀ = v1 = v1 , π P = Pv1 · · · vn , π f = f v1 · · · vn = vn+1 . The conditions (i) and (ii) are then met no matter what T1 is. Now assume that π is an interpretation and let B be a model of T1 . There is a natural way to extract from B a structure π B for L 0 . Namely, let |π B| = the set deﬁned in B by π∀ , π P B = the relation deﬁned in B by π P , restricted to |π B|, π B f (a1 , . . . , an ) = the unique b such that |=B π f [[a1 , . . . , an , b]], where a1 , . . . , an are in |π B|.

∅. And, by By condition (i) in the deﬁnition of interpretations, |π B| = condition (ii), the deﬁnition of f π B makes sense; i.e., there is a unique b meeting the above condition. Hence π B is indeed a structure for the language L 0 . Deﬁne the set π −1 [T1 ] of L 0 -sentences by the equation π −1 [T1 ] = Th{π B | B ∈ Mod T1 } = {σ | σ is an L 0 -sentence true in every structure π B obtainable from a model B of T1 }.

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This is a theory, as is Th K for any class K. It is a satisﬁable theory iff T1 is satisﬁable. EXAMPLE. Earlier in this section we had a theory T containing the sentence ∀ v1 ∃!v2 ϕ.

(ε)

We augmented the language to a larger language L + that contained a function symbol f . The “deﬁnition” of f was provided by the L + -sentence ∀ v1 ∀ v2 ( f v1 = v2 ↔ ϕ).

(δ)

We showed that for a sentence σ in the original language of T , if T ; δ |= σ , then T |= σ . We have an interpretation π from L + into T . π is the identity interpretation on all parameters except f . The formula π f is ϕ. The fact that T |= ε is just what we need to verify that π is indeed an interpretation. For any model A of T , π A is a structure previously called (A, F); it is a model of T ; δ. We claim that π −1 [T ] = Cn(T ; δ). First observe that any model B of T ; δ equals π B, where A is the restriction of B to the language of T . Hence for an L + -sentence σ , σ ∈ π −1 [T ] ⇔ |=π A σ for every model A of T for every model B of T ; δ ⇔ |=B σ ⇔ T ; δ |= σ.

Syntactical Translation In the preceding subsection on interpretations we talked about arbitrary models and such. But the reader may already have noticed that there is a much more down-to-earth thing to be said about an interpretation π of L 0 into T1 . Brieﬂy: We can, given a formula ϕ of L 0 , ﬁnd a formula ϕ π in L 1 which in some sense corresponds exactly to ϕ. We deﬁne ϕ π by recursion on ϕ. First, consider an atomic formula α of L 0 . For example, if α is P f gx, then α is logically equivalent to ∀ y(gx = y → ∀ z( f y = z → P z)). And we can take for α π the L 1 -formula ∀ y(πg (x, y) → ∀ z(π f (y, z) → π P (z))).

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A Mathematical Introduction to Logic In general, scan an atomic formula α from right to left. The rightmost place at which a function symbol occurs will initiate a segment of the form gx1 · · · xn , for some n-place g. (In the example n = 1.) Replace this by some new variable y, and preﬁx ∀ y(πg (x1 , . . . , xn , y))→. Continue to the next place at which a function symbol occurs. Finally, replace the predicate symbol P (if a parameter) by π P (with the correct variables). The deﬁnition of α π can be stated more carefully by using recursion on the number of places at which function symbols occur in α. If that number is zero, then α is P x1 · · · xn and α π is π P (x1 , . . . , xn ). Otherwise, take the rightmost place at which a function symbol g occurs. If g is an n-place symbol, then that place initiates a segment gx1 · · · xn . Replace this segment by some new variable y, obtaining a formula we can call α ygx1 ···xn . Then α π is ∀ y(πg (x1 , . . . , xn , y) → (α ygx1 ···xn )π ). For example, (P f gx)π = ∀ y(πg (x, y) → (P f y)π ) = ∀ y(πg (x, y) → ∀ z(π f (y, z) → (P z)π )) = ∀ y(πg (x, y) → ∀ z(π f (y, z) → π P (z))). The interpretation of a nonatomic formula is deﬁned in the obvious way. (¬ ϕ)π is (¬ ϕ π ), (ϕ → ψ)π is (ϕ π → ψ π ), and ( ∀ x ϕ)π is ∀ x(π∀ (x) → ϕ π ). (Thus the quantiﬁers are “relativized” to π∀ .) The sense in which ϕ π “says the same thing” as ϕ is made precise in the following basic lemma. LEMMA 27B Let π be an interpretation of L 0 into T1 , and let B be a model of T1 . For any formula ϕ of L 0 and any map s of the variables into |π B|, |=π B ϕ[s]

iff |=B ϕ π [s].

This is not a deep fact. It just says that ϕ π was deﬁned correctly. PROOF. We use induction on ϕ, but only the case of an atomic formula α is nontrivial. For α, we use induction on the number of places at which function symbols occur. It is easy if that number is zero. Otherwise, α π = ∀ y(πg (x, y) → β π ), y = α. (We have quietly assumed that g is a one-place where βgx symbol; the notation is bad enough already.) Let

b = the unique b such that |=B πg [[s(x), b]] π = g B (s(x)).

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Then |=B α π [s] ⇔ |=B β π [s(y | b)] ⇔ |=π B β[s(y | b)] by the inductive hypothesis y ⇔ |=π B βgx [s] by the substitution lemma ⇔ |=π B α[s].

The following corollary justiﬁes our choice of notation for π −1 [T1 ]. COROLLARY 27C

For a sentence σ of L 0 , σ ∈ π −1 [T1 ]

PROOF.

iff

σ π ∈ T1 .

Recall that by deﬁnition

σ ∈ π −1 [T1 ] ⇔ for every model B of T1 , |=π B σ ⇔ for every model B of T1 , |=B σ π by Lemma 27B ⇔ T1 |= σ π .

DEFINITION. An interpretation π of a theory T0 into a theory T1 is an interpretation π of the language of T0 into T1 such that T0 ⊆ π −1 [T1 ]. In other words, it is necessary that for an L 0 -sentence σ , σ ∈ T0 ⇒ σ π ∈ T1 . π −1 [T1 ] is the largest theory that π interprets into T1 . If T0 = π −1 [T1 ], then we have σ ∈ T0 ⇔ σ π ∈ T1 . In this case π is said to be a faithful interpretation of T0 into T1 . To return to an earlier example, consider the structures (N; 0, S) and (Z; +, ·). We had an interpretation π into Th(Z; +, ·), where π∀ (x) = ∃ y1 ∃ y2 ∃ y3 ∃ y4 x = y1 · y1 + y2 · y2 + y3 · y3 + y4 · y4 , π0 (x) = x + x = x, z → x + z = y). πS (x, y) = ∀ z(z · z = z ∧ z + z = We now claim that π is a faithful interpretation of Th(N; 0, S) into Th(Z; +, ·). For in this case, π (Z; +, ·) is the structure (N; 0, S). Hence |=(N;0,S) σ ⇐⇒ |= π(Z;+,·) σ ⇐⇒ |=(Z;+,·) σ π . In Chapter 3 we will be able to show that there is no interpretation of Th(Z; +, ·) into Th(N; 0, S). Thus the former theory is strictly stronger than the latter.

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A Mathematical Introduction to Logic Finally, let us return to the situation with which we started this section. Assume that T is a theory containing the sentence ε, where ε = ∀ v1 ∃!v2 ϕ; δ = ∀ v1 ∀ v2 ( f v1 = v2 ↔ ϕ); L + = the language obtained by adding the new function symbol f to the language of T ; π = the interpretation of L + into T that is the identity interpretation on all parameters except f , and π f = ϕ. In fact, π is a faithful interpretation of Cn(T ; δ) into T , since, as noted previously, π −1 [T ] = Cn(T ; δ). We can now draw an additional conclusion; the deﬁnition is eliminable. THEOREM 27D Assume that we have the situation described above. Then for any L + -sentence σ we can ﬁnd the sentence σ π in the original language such that (a) T ; δ |= (σ ↔ σ π ). (b) T ; δ |= σ ⇐⇒ T |= σ π . (c) If f does not occur in σ , then |= (σ ↔ σ π ). PROOF. Part (c) follows from the fact that π is the identity interpretation on all parameters except f . Part (b) restates that π is a faithful interpretation of Cn(T ; δ) into T . Since π is faithful, for (a) it sufﬁces to show that T |= (σ ↔ σ π )π . This follows from (c), since (σ ↔ σ π )π is (σ π ↔ σ π π ), which is valid.

Exercises 1. Assume that L 0 and L 1 are languages with the same parameters except that L 0 has an n-place function symbol f not in L 1 and L 1 has an (n + 1)-place predicate symbol P not in L 0 . Show that for any L 0 -theory T there is a faithful interpretation of T into some L 1 -theory. 2. Let L 0 be the language with equality and the two-place function symbols + and ·. Let L 1 be the same, but with three-place predicate symbols for addition and multiplication. Let Ni = (N; +, ·) be the structure for L i consisting of the natural numbers with addition and multiplication (i = 0, 1). Show that any relation deﬁnable by an L 0 -formula in N0 is also deﬁnable by an L 1 -formula in N1 .

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3. Show that an interpretation of a complete theory into a satisﬁable theory is faithful.

SECTION 2.8 Nonstandard Analysis1 The differential and integral calculus was originally described by Leibniz and Newton in the seventeenth century in terms of quantities that were inﬁnitely small yet nonzero. Newton used in his calculations a number o that, being inﬁnitely small, could be multiplied by any ﬁnite number and still be negligible. But it was necessary to divide by o, so it had to be nonzero. Leibniz’s d x was less than any assignable quantity, yet was nonzero. These ideas were not easy to comprehend or to accept. Throughout the eighteenth century this business of working with inﬁnitesimals was attacked (e.g., by Bishop Berkeley), distrusted (e.g., by D’Alembert), and used in enthusiastic experimentation (e.g., by Euler). While Euler was creating the mathematics that students now study in advanced calculus, he used inﬁnitesimals in a loose, free-swinging manner that would not be tolerated in today’s freshmen. Only in the nineteenth century were the foundations of calculus presented in the form now found in textbooks. The treatment of limits was then rigorous, and debate subsided. In 1961 Abraham Robinson introduced a new method for treating limits, rescuing inﬁnitesimals from their intellectual disrepute. This method combines the intuitive advantages of working with inﬁnitely small quantities with modern standards of rigor. The basic idea is to utilize a nonstandard model of the theory of the real numbers.

Construction of ∗ R We will use a very large ﬁrst-order language. In addition to symbols for +, ·, and < we might as well add symbols for the exponentiation and absolute-value functions. And since there is no good reason to stop there, we go all the way and include a symbol for every operation on the set R of reals. We do the same for every relation on R. Thus we have the language with equality and the following parameters: 0. 1. 2. 3.

1

∀, intended to mean “for all real numbers.” An n-place predicate symbol P R for each n-ary relation R on R. A constant symbol cr for each r ∈ R. An n-place function symbol f F for each n-ary operation F on R.

This section may be omitted without loss of continuity.

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A Mathematical Introduction to Logic For this language there is the standard structure R, with |R| = R, R R PR R = R, cr = r , and f F = F. But now let us form a nonstandard structure, by using the compactness theorem. Let be the set Th R ∪ {cr P< v1 | r ∈ R}. (Here cr P< v1 formalizes “r is less than v1 .”) Any ﬁnite subset of can be satisﬁed in R by assigning to v1 some large real number. Hence by the compactness theorem there is a structure A and an element a ∈ |A| such that is satisﬁed in A when v1 is assigned a. Since A is a model of Th R, we have A ≡ R. There is also an isomorphism h of R into (but not onto) A deﬁned by h(r ) = crA . To check that this is indeed an isomorphism, we use the fact that A ≡ R.

r2 , the sentence cr1 = cr2 holds in R and h is one-to-one, since for r1 = hence in A. h preserves a binary relation R(= PR R ) since for any r and s in R, r, s ∈ PR R ⇔ |=R P R cr cs ⇔ |= A P R cr cs A ⇔ crA , cA s ∈ PR ⇔ h(r ), h(s) ∈ PAR . A similar calculation applies to any n-ary relation. Next we must show that h preserves any function F(= f R F ). Again for notational ease, suppose that F is a binary operation. Consider any r and s in R, and let t = F(r, s). Then

h fR F (r, s) = h(F(r, s)) = h(t) = cA t . Now the sentence ct = f F cr cs holds in R and hence in A. Hence

A A A cA t = f F cr , cs = f AF (h(r ), h(s)). So h preserves f F . For the constant symbols we have by the deﬁnition of h, h crR = h(r ) = crA . Since we have an isomorphic copy of R inside A, we can ﬁnd another structure ∗ R isomorphic to A such that R is a substructure of ∗ R. The idea is simply to replace in A the point crA by the point r (provided that |A| ∩ R = ∅, as can always be arranged). For details, see Exercise 24 of Section 2.2. Since ∗ R is isomorphic to A, there is a point b ∈ |∗ R| such that ∗ R satisﬁes when v1 is assigned b. In particular, ∗ R ≡ R.

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To go much further, we need a less cumbersome notation. We will use an asterisk to indicate passage from R to ∗ R. ∗

1. For each n-ary relation R on R, let ∗ R be the relation P RR assigned to the symbol P R by ∗ R. In particular, R is a unary relation on R. Its image ∗ R equals the universe of ∗ R, since the sentence ∀ xPR x is true in R and hence in ∗ R. Since R is a substructure of ∗ R, we have that each relation R equals the restriction of ∗ R to R. ∗ 2. For each n-ary operation F on R, let ∗ F be the operation f RR assigned to the symbol f F by ∗ R. F is then the restriction to R of ∗ F. ∗

Observe that cr R = r , so we need no special notation for this. There is a general method (to be used heavily in the remainder of this section) for demonstrating properties of a relation ∗ R or an operation ∗ F. One simply observes (1) that R or F has the property, (2) that the property can be expressed by a sentence of the language, and (3) that R ≡ ∗ R. For example, the binary relation ∗< on ∗ R is transitive. This is because < is transitive, and this property can be expressed by the sentence ∀ x ∀ y ∀ z(xP< y → yP< z → xP< z). By similar reasoning, ∗ < satisﬁes trichotomy on ∗ R and thus is an ordering relation on ∗ R. For another example, we can prove that the binary operation ∗ + on ∗ R is commutative, since + is commutative and the commutative law can be expressed by a sentence. By applying this reasoning to each of the ﬁeld axioms, we see that (∗ R; 0, 1, ∗ +, ∗ ·) is a ﬁeld. This general method is used so much, we will shortly begin to take it for granted. If, for example, we assert that ∗ |a ∗ + b| ∗≤ ∗ |a| ∗ + ∗ |b| for a and b in ∗ R, we will take for granted that the reader perceives that the general method yields this fact. We have R ⊆ ∗ R, but R =

∗ R. For we have some point b such that ∗ |=∗ R cr P< v1 [[b]]; i.e., r < b. Thus b is inﬁnitely large, being larger (in the ordering ∗ <) than any standard r , i.e., any r ∈ R. Its reciprocal 1 ∗ / b will be a sample inﬁnitesimal. Properties of R that cannot be expressed in the language are likely to fail in ∗ R. The least-upper-bound property is one such. There are non-empty bounded subsets S of ∗ R that have no least upper bound (with respect to the ordering ∗ <). For example, R is such a subset of ∗ R. It is bounded by the inﬁnite b of the preceding paragraph. But it has no least upper bound; see Exercise 7. Deﬁne the set F of ﬁnite elements by the equation F = {x ∈ ∗ R | ∗ |x| ∗< y

for some

y ∈ R}.

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A Mathematical Introduction to Logic Similarly, deﬁne the set I of inﬁnitesimals by the equation I = {x ∈ ∗ R | ∗ |x| ∗< y

for all positive

y ∈ R}.

If A ⊆ R is unbounded, then ∗ A contains inﬁnite points. For the sentence “for any real r there is an element a ∈ A larger than r ” is true and formalizable. Take some inﬁnite positive b; there must be a larger (and hence inﬁnite) member of ∗ A. For example, ∗ N contains inﬁnite numbers. The only standard inﬁnitesimal, i.e., the only member of R ∩ I, is 0. But there are other inﬁnitesimals. For by the usual (formalizable) rules for inequalities, the reciprocal of any inﬁnite number is an inﬁnitesimal.

Algebraic Properties In the next theorem we collect some algebraic facts about F and I that will be useful later. THEOREM 28A (a) F is closed under addition ∗ +, subtraction ∗ −, and multiplication ∗ ·. (b) I is closed under addition ∗ +, subtraction ∗ −, and multiplication from F: x ∈I

and

z ∈ F ⇒ x ∗ · z ∈ I.

In algebraic terminology, part (a) says that F is a subring of the ﬁeld R, and part (b) says that I is an ideal in the ring F. We will see a little later what the quotient ring F/I is. ∗

PROOF. (a) Let x and y be ﬁnite, so that ∗ |x| ∗< a, ∗ |y| ∗< b for standard a and b in R. Then ∗

|x ∗ ± y| ∗≤ ∗ |x| ∗ + ∗ |y| ∗< a + b ∈ R,

whence x ∗ + y, x ∗ − y are ﬁnite. Also ∗

|x ∗ · y| ∗< a · b ∈ R,

whence x ∗ · y is ﬁnite, too. (b) Let x and y be inﬁnitesimals. Then for any positive standard a, ∗ |x| ∗< a/2 and ∗ |y| ∗< a/2. Hence ∗

|x ∗ ± y| ∗< a/2 + a/2 = a,

so that x ∗ + y and x ∗ − y are inﬁnitesimal. If z is ﬁnite, then |z| ∗< b for some standard b. Since x is inﬁnitesimal, we have ∗ |x| ∗< a/b, whence ∗

∗

|x ∗ · z| ∗< (a/b)b = a.

Thus x ∗ · z is also inﬁnitesimal.

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DEFINITION. x is inﬁnitely close to y (written x # y) iff x ∗ − y is inﬁnitesimal. THEOREM 28B (a) # is an equivalence relation on ∗ R. (b) If u # v and x # y, then u ∗ + x # v ∗ + y and ∗ − u # ∗ − v. (c) If u # v and x # y and x, y, u, v are ﬁnite, then u ∗ · x # ∗ v · y. PROOF. This is a consequence of part (b) of the preceding theorem (I is an ideal in F). (a) # is reﬂexive since 0 is inﬁnitesimal. # is symmetric since the negative (∗ −) of an inﬁnitesimal is inﬁnitesimal. Finally, suppose that x # y and y # z. Then x ∗ − z = (x ∗ − y) ∗ + (y ∗ − z) ∈ I since I is closed under addition. (b) If u # v and x # y, then (u ∗ + x) ∗ − (v ∗ + y) = (u ∗ − v) ∗ + (x ∗ − y) ∈ I since I is closed under addition. Also ∗ − u # ∗ − v since I is closed under negation. (c) (u ∗ · x) ∗ − (v ∗ · y) = (u ∗ · x) ∗ − (u ∗ · y) ∗ + (u ∗ · y) ∗ − (v ∗ · y) = u ∗ · (x ∗ − y) ∗ + (u ∗ − v) ∗ · y ∈ I since I is closed under multiplication from F.

For standard r and s, we have r # s iff r = s, as 0 is the only standard inﬁnitesimal. LEMMA 28C If x # y and at least one is ﬁnite, then there is a standard q strictly between x and y. PROOF. We may suppose that x ∗< y. In fact, we may further suppose that 0 ∗< x ∗< y; the case x ∗< y ∗≤ 0 is similar and the case x ∗< 0 ∗< y is trivial. Since x # y there is a standard b such that 0 < b ∗< y ∗ − x. Since x is ﬁnite we have x ∗< mb for some positive integer m; take the least such m. Then x ∗< mb ∗< y. (By the leastness of m, (m − 1)b ∗≤ x. So mb ∗≤ x ∗ + b ∗< y.) THEOREM 28D PROOF.

Every x ∈ F is inﬁnitely close to a unique r ∈ R.

For x ∈ F the set S = {y ∈ R | y ∗< x}

of standard points below x has an upper bound in R. Let r be its least upper bound; we claim that x # r .

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A Mathematical Introduction to Logic If x # r , then by the lemma there is a standard q between x and r . If r < q ∗< x, then r fails to be an upper bound for S. If x ∗< q < r , then q is also an upper bound for S, contradicting the leastness of r . Hence x # r . This establishes the existence of r . As for the uniqueness, note that if x # r and x # s, then r # s. For standard r and s this implies that r = s. COROLLARY 28E Each ﬁnite x has a unique decomposition of the form x = s ∗ + i, where s is standard and i is inﬁnitesimal. We call s the standard part of x, written st(x). (Another notation for the standard part of x is ◦ x.) Of course for standard r , st(r ) = r . The next theorem summarizes some properties of the st function. THEOREM 28F (a) st maps F onto R. (b) st(x) = 0 iff x is inﬁnitesimal. (c) st(x ∗ + y) = st(x) + st(y). (d) st(x ∗ · y) = st(x) · st(y). PROOF. (a) and (b) are clear. Since st(x) # x and st(y) # y, we have by part (b) of Theorem 28B that st(x) + st(y) # x ∗ + y. Hence the left side equals st(x ∗ + y). Part (d) is similar and uses part (c) of Theorem 28B. (In algebraic terminology, this theorem asserts that st is a homomorphism of the ring F onto the ﬁeld R, with kernel I. Consequently, the quotient ring F/I is isomorphic to the real ﬁeld R.) Henceforth in this section we will streamline our notation by omitting the asterisks on the symbols for the arithmetic operations ∗ +, ∗ −, ∗ ·, and ∗ /.

Convergence In calculus courses convergence is usually treated in terms of ε’s and δ’s and variables that come delicately close to certain values. We will give here the beginning of an alternative treatment of convergence, where variables come inﬁnitely close to the limiting values. DEFINITION. Let F : R → R. Then F converges at a to b iff whenever x is inﬁnitely close to (but different from) a, then ∗ F(x) is inﬁnitely close to b. PROOF OF EQUIVALENCE WITH THE ORDINARY DEFINITION. First suppose that F converges at a to b in the ordinary sense. That is, for any ε > 0 there is a δ > 0 such that 0=

|x − a| < δ ⇒ |b − F(x)| < ε

for any x.

The displayed sentence (concerning the standard numbers ε and

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δ) is formalizable and thus holds in ∗ R. Now if x in ∗ R is inﬁnitely close to (but different from) a, then certainly 0 =

∗ |x − a| ∗< δ. ∗ ∗ ∗ Hence |b − F(x)| < ε. Since ε was arbitrary, b # ∗ F(x). Conversely, assume that the condition stated in the deﬁnition is met. Then for any standard ε > 0, the sentence There exists δ > 0 such that for all x, 0 =

|a − x| < δ ⇒ |b− F(x)| < ε (when formalized) holds in ∗ R, since we can take δ to be inﬁnitesimal. Hence the sentence holds in R also. First remark: It is entirely possible that F does not converge at a to any number. On the other hand, F converges at a to at most one b. For if i is a nonzero inﬁnitesimal, then b = st(∗ F(a + i)). It is traditional to denote this b by “limx→a F(x).” Thus lim F(x) = st(∗ F(a + i)).

x→a

Second remark: It is not really necessary to have dom F = R. It is enough for a to be an accumulation point of dom F. (a is an accumulation point of S iff a is inﬁnitely close to, but different from, some member of ∗ S.) COROLLARY 28G F is continuous at a iff whenever x # a, then ∗ F(x) # F(a). Now consider a function F : R → R and a standard a ∈ R. Then the derivative F (a) is F(a + h) − F(a) h→0 h By our deﬁnition of limit, this can also be stated: F (a) = b iff for every nonzero inﬁnitesimal d x we have d F/d x # b, where d F = ∗ F(a + d x) − F(a). Thus if there is such a b (i.e., if F (a) exists), then lim

F (a) = st(d F/d x) for any nonzero inﬁnitesimal d x. Here d F/d x is the result of dividing d F by d x. The fact that we simply use division here greatly facilitates calculations. EXAMPLE.

Let F(x) = x 2 . Then F (a) = 2a, since

dF (a + d x)2 − a 2 2a(d x) + (d x)2 = = = 2a + d x # 2a. dx dx dx THEOREM 28H PROOF.

If F (a) exists, then F is continuous at a.

For any nonzero inﬁnitesimal d x we have ∗

F(a + d x) − F(a) # F (a). dx

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A Mathematical Introduction to Logic The right side is standard, so the left side is at least ﬁnite. Consequently, when we multiply the left side by the inﬁnitesimal d x we are left with the fact that ∗ F(a + d x) − F(a) ∈ I; i.e., ∗ F(a + d x) # F(a). The reader should note that this result is not some nonstandard analogue of a classical theorem, nor even a generalization of a classical theorem. It is a classical theorem. It is only the proof that is nonstandard. The same remarks apply to the next theorem. Let F ◦ G be the function whose value at a is F(G(a)). CHAIN RULE Assume that G (a) and F (G(a)) exist. Then (F ◦ G) (a) exists and equals F (G(a)) · G (a). PROOF. First, notice that ∗ (F ◦ G) =∗ F ◦ ∗ G, since the sentence ∀ v1 f F◦G v1 = f F f G v1 holds in the structures. Now consider any nonzero inﬁnitesimal d x. Let dG = ∗ G(a + d x) − G(a), d F = ∗ (F ◦ G)(a + d x) − (F ◦ G)(a) = ∗ F(∗ G(a + d x)) − F(G(a)) = ∗ F(G(a) + dG) − F(G(a)). Then dG # 0 since G is continuous at a. If dG =

0, then by the last of these equations d F/dG # F (G(a)), whence dF d F dG = · # F (G(a)) · G (a). dx dG d x If dG = 0, then d F = 0 and G (a) # dG/d x = 0, so we again have dF # F (G(a)) · G (a). dx These theorems are but samples of the treatment of convergence in terms of inﬁnite proximity. The method is not at all limited to elementary ∞ topics. One can construct delta functions δ with the property that −∞ δ = 1 and yet δ(x) # 0 for x # 0. Original results in analysis (e.g., in the theory of Hilbert spaces) have been obtained by the method of nonstandard analysis. Possibly the method will be more widely used in the future as more analysts become familiar with it.

Exercises 1. (Q is dense in R.) Let Q be the set of rational numbers. Show that every member of ∗ R is inﬁnitely close to some member of ∗ Q. 2. (a) Let A ⊆ R and F : A → R. Then F is also a binary relation on R; show that ∗ F : ∗ A → ∗ R. (b) Let S : N → R. Recall that S is said to converge to b iff for every ε > 0 there is some k such that for all n > k, |S(n) − b| < ε.

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Show that this is equivalent to the condition: ∗ S(x) # b for every inﬁnite x ∈ ∗ N. (c) Assume that Si : N → R and Si converges to bi for i = 1, 2. Show that S1 + S2 converges to b1 + b2 and S1 · S2 converges to b1 · b2 . 3. Let F : A → R be one-to-one, where A ⊆ R. Show that if x ∈ ∗ A / R. but x ∈ / A, then ∗ F(x) ∈ 4. Let A ⊆ R. Show that A = ∗ A iff A is ﬁnite. 5. (Bolzano–Weierstrass theorem) Let A ⊆ R be bounded and inﬁnite. Show that there is a point p ∈ R that is inﬁnitely close to, but different from, some member of ∗ A. Suggestion: Let S : N → A with S one-to-one; look at ∗ S(x) for inﬁnite x ∈ ∗ N. 6. (a) Show that ∗ Q has cardinality at least 2ℵ0 , where Q is the set of rational numbers. Suggestion: Use Exercise 1. (b) Show that ∗ N has cardinality at least 2ℵ0 . 7. Let A be a subset of R having no greatest member. Then as a subset of ∗ R, A will have upper bounds (with respect to the ordering ∗ <) in ∗ R. But show that A does not have a least such bound.

Chapter

T H R E E

Undecidability SECTION 3.0 Number Theory In this chapter we will focus our attention on a speciﬁc language, the language of number theory. This will be the ﬁrst-order language with equality and with the following parameters: ∀, intended to mean “for all natural numbers.” (Recall that the set N of natural numbers is the set {0, 1, 2, . . .}. Zero is natural.) 0, a constant symbol intended to denote the number 0. S, a one-place function symbol intended to denote the successor function S : N → N, i.e., the function for which S(n) = n + 1. <, a two-place predicate symbol intended to denote the usual (strict) ordering relation on N. +, ·, E, two-place function symbols intended to denote the operations +, ·, and E of addition, multiplication, and exponentiation, respectively. We will let N be the intended structure for this language. Thus we may informally write N = (N; 0, S, <, +, ·, E). (More precisely, |N| = N, 0N = 0, and so forth.) By number theory we mean the theory of this structure, Th N. As warmup exercises we will study (in Sections 3.1 and 3.2) certain reducts of N, 182

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i.e., restrictions of N to sublanguages: N S = (N; 0, S), N L = (N; 0, S, <), N A = (N; 0, S, <, +). Finally, in Section 3.8 we will consider N M = (N; 0, S, <, +, ·). For each of these structures we will raise the same questions: (A) Is the theory of the structure decidable? If so, what is a nice set of axioms for the theory? Is there a ﬁnite set of axioms? (B) What subsets of N are deﬁnable in the structure? (C) What do the nonstandard models of the theory of the structure look like? (By “nonstandard” we mean “not isomorphic to the intended structure.”) Our reason for choosing number theory (rather than, say, group theory) for special study is this: We can show that a certain subtheory of number theory is an undecidable set of sentences. We will also be able to infer that any satisﬁable theory that is at least as strong as this fragment of number theory (e.g., the full number theory or set theory) must be undecidable. In particular, such a theory cannot be both complete and axiomatizable. In order to show that our subtheory of number theory is undecidable, we will show that it is strong enough to represent (in a sense to be made precise) facts about sequences of numbers, certain operations on numbers, and ultimately facts about decision procedures. This last feature then lets us perform a diagonal argument that demonstrates undecidability. We could alternatively use, in place of a subtheory of number theory, some other theory (such as a fragment of the theory of ﬁnite sets) in which we could conveniently represent facts about decision procedures. Before giving examples of the expressiveness of the language of number theory, it is convenient to introduce some notational conventions. As a concession to everyday usage, we will write x < y,

x + y,

x · y,

+ x y,

· x y,

and

xEy

in place of the ofﬁcial < x y,

and

E x y.

For each natural number k we have a term Sk 0 (the numeral for k) that denotes it: S0 0 = 0,

S1 0 = S0,

S2 0 = SS0,

etc.

(The set of numerals is generated from {0} by the operation of

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A Mathematical Introduction to Logic preﬁxing S.) The fact that every natural number can be named in the language will be a useful feature. Even though only countably many relations on N are deﬁnable in N, almost all the familiar relations are deﬁnable. For example, the set of primes is deﬁned in N by S1 0 ∧ ∀ v2 ∀ v3 (v1 = v2 · v3 → v2 = S1 0 ∨ v3 = S1 0). v1 = Later we will ﬁnd it important to show that many other speciﬁc relations are deﬁnable in N. One naturally expects the expressiveness of the language to be severely restricted when some of the parameters are omitted. For example, the set of primes, as we shall see, is not deﬁnable in N A . On the other hand, in Section 3.8 we will show that any relation deﬁnable in N is also deﬁnable in N M .

Preview The main theorems of this chapter — the theorems associated with the names of G¨odel, Tarski, and Church — are proved in Section 3.5. But we can already sketch here some of the ideas involved. We want to compare the concepts of truth and proof ; that is, we want to compare the set of sentences true in N with the set of sentences that might be provable from an appropriate set A of axioms. We can assign to each formula α of the language of number theory an integer α, called the G¨odel number of α. Any sufﬁciently straightforward way of assigning distinct integers to formulas would sufﬁce for our purposes; a particular assignment is adopted at the beginning of Section 3.4. What is important is that from α we can effectively ﬁnd the number α, and conversely. Similarly, to each ﬁnite sequence D of formulas (such as a deduction) we assign an integer G(D). Note that for any set A of formulas, we can form the corresponding set {α | α ∈ A} of numbers. There are now three ways in which to proceed: the self-reference approach, the diagonalization approach, and the computability approach. It will be argued later, however, that the three approaches are more closely related than they appear — they are three aspects of one approach. First, in the self-reference approach, we make a sentence σ that can be thought of as saying, “I am unprovable.” More speciﬁcally, we have the following: THEOREM 30A Let A ⊆ Th N be a set of sentences true in N, and assume that the set {α | α ∈ A} of G¨odel numbers of members of A is a set deﬁnable in N. Then we can ﬁnd a sentence σ such that σ is true in N but σ is not deducible from A.

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PROOF. We will construct σ to express (in an indirect way) that σ itself is not a theorem of A. Then the argument will go roughly as follows: If A ! σ , then what σ says is false, contradicting the fact that A consists of true sentences. And so A σ , whence σ is true. To construct σ , we begin by considering the ternary relation R deﬁned by a, b, c ∈ R iff a is the G¨odel number of some formula α and c is the value of G at some deduction from A of α(Sb 0). Then because {α | α ∈ A} is deﬁnable in N, it follows that R is deﬁnable also. (The details of this step must wait until later sections.) Let ρ be a formula that deﬁnes R in N. Let q be the G¨odel number of ∀ v3 ¬ ρ(v1 , v1 , v3 ). (We use here the notation: ϕ(t) = ϕtv1 , ϕ(t1 , t2 ) = (ϕtv1 1 )vt22 , and so forth.) Then let σ be ∀ v3 ¬ ρ(Sq 0, Sq 0, v3 ). Thus σ says that no number is the value of G at a deduction from A of the result of replacing, in formula number q, the variable v1 by the numeral for q; i.e., no number is the value of G at a deduction of σ . Suppose that, contrary to our expectations, there is a deduction of σ from A. Let k be the value of G at a deduction. Then q, q, k ∈ R and hence |=N ρ(Sq 0, Sq 0, Sk 0). It is clear that σ ! ¬ ρ(Sq 0, Sq 0, Sk 0) and the two displayed lines tell us that σ is false in N. But A ! σ and the members of A are true in N, so we have a contradiction. Hence there is no deduction of σ from A. And so for every k, we have q, q, k ∈ / R. Thus for every k |=N ¬ ρ(Sq 0, Sq 0, Sk 0), from which it follows (with the help of the substitution lemma) that |=N ∀ v3 ¬ ρ(Sq 0, Sq 0, v3 ); i.e., σ is true in N.

We will argue later — using something called Church’s thesis — that any decidable set of natural numbers must be deﬁnable in N. The conclusion will then be that Th N is not axiomatizable.

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A Mathematical Introduction to Logic COROLLARY 30B The set {τ | |=N τ } of G¨odel numbers of sentences true in N is a set that is not deﬁnable in N. PROOF. If this set were deﬁnable, we could take A = Th N in the preceding theorem to obtain a contradiction. Section 3.5 will follow the self-reference approach, but with a variation in which the sentence σ tries to say, “I am false.” (The well-known liar paradox is relevant here!) But if this “self-reference” construction seems too much like a magic trick, there is a second way to describe the situation: the diagonalization approach, which does not use an obvious self-reference. We start by deﬁning the following binary relation P on the natural numbers: a, b ∈ P ⇐⇒ a is the G¨odel number of a formula α(v1 ) (with just v1 free) and |=N α(Sb 0). (More informally, a, b ∈ P ⇔ “a is true of b.”) Then any set of natural numbers that is deﬁnable in N equals, for some a, the “vertical section” Pa = {b | a, b ∈ P} of P. Namely, we take a to be the G¨odel number of a formula deﬁning the set, and use the fact that |=N α(Sb 0) ⇔ |=N α(v1 )[[b]]. So any deﬁnable (in N) set of natural numbers is somewhere on the list P1 , P2 , . . . . Now we “diagonalize out” of the list. Deﬁne the set: H = {b | b, b ∈ / P}. (More informally, b ∈ H ⇔ “b is not true of b.”) Then H is nowhere on the list P1 , P2 , . . . . (H = P3 because 3 ∈ H ⇔ 3 ∈ / P3 , so the number 3 belongs to exactly one of these two sets and not to the other.) Therefore H is not deﬁnable in N. Why is H undeﬁnable? After all, we have above speciﬁed that b ∈ H ⇐⇒ not [b is not the G¨odel number of a formula α(v1 ) (with just v1 free) and |=N α(Sb 0)]. What is the barrier to translating this speciﬁcation into the language of arithmetic? We will show that the barrier is not being the G¨odel number of a formula — we can translate that — and the barrier is not having v1 free and not substituting the numeral Sb 0 into a formula. By the process of elimination, we will show that the only possible barrier is saying of a sentence that is true in N. THEOREM 30C (a) The set {τ | |=N τ } of G¨odel numbers of sentences true in N is not deﬁnable in N.

(b) The theory Th N is undecidable.

(c) The theory Th N is not axiomatizable.

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PROOF. Part (a), which is the same as Corollary 30B in the selfreference approach, has the diagonal proof sketched above. That is, if to the contrary Th N were deﬁnable in N, then the above set H would also be deﬁnable, which it is not. Part (b) will then follow, after we argue every decidable set of natural numbers must be deﬁnable in N. If Th N were decidable, then the corresponding set {τ | |=N τ } of numbers would be decidable and hence deﬁnable in N, which it is not. And part (c) is an immediate consequence of part (b) and Corollary 26I, since Th N is a complete theory. And thirdly, the computability approach presents us with a stark difference between what is true and what is provable. From Section 2.6 we know that whenever A is a decidable set (or even an effectively enumerable set) of axioms we might choose for Th N, the set Cn A of provable sentence will be an effectively enumerable set. In contrast, the computability approach will show — using Church’s thesis — that the set Th N of all true sentences is not effectively enumerable. This fact, which is closely related to Theorem 30C, will follow from another diagonal argument in Section 3.6.

THEOREM 30D For any decidable (or even effectively enumerable) set A of axioms, Cn A = Th N because the set on the left is effectively enumerable and the set on the right is not.

Theorem 30D presents the dilemma: Either the axioms are lying to us by allowing us to deduce false sentences, or else the axioms are incomplete, in the sense that some true sentences cannot be deduced from those axioms. This computability approach is implicit in parts of Section 3.5, but it is in Section 3.6 that the approach explicitly appears, and where it is compared with the other two approaches.

SECTION 3.1 Natural Numbers with Successor We begin with a situation that is simple enough to let us give reasonably complete answers to our questions. We reduce the set of parameters to just ∀, 0, and S, eliminating <, +, ·, and E. The corresponding reduct of N is N S = (N; 0, S).

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A Mathematical Introduction to Logic In this restricted language we still have the numerals, naming each point in N. But the sentences we can express in the language are, from the viewpoint of arithmetic, uninteresting. We want to ask about N S the same questions that interest us in the case of N. We want to know about the complexity of the set Th N S ; we want to study deﬁnability in N S ; and we want to survey the nonstandard models of N S . To study the theory of the natural numbers with successor (Th N S ), we begin by listing a few of its members, i.e., sentences true in N S . (These sentences will ultimately provide an axiomatization for the theory.) S1. ∀ x Sx = 0, a sentence asserting that zero has no predecessor. S2. ∀ x ∀ y(Sx = Sy → x = y). This asserts that the successor function is one-to-one. S3. ∀ y(y = 0→∃ x y = Sx). This asserts that any nonzero number is the successor of something. S4.1 ∀ x Sx = x. S4.2 ∀ x SSx = x. ··· S4.n ∀ x Sn x = x, where the superscript n indicates that the symbol S occurs at n consecutive places. Let A S be the set consisting of the above sentences S1, S2, S3, S4.n (n = 1, 2, . . .). Clearly these sentences are true in N S ; i.e., N S is a model of A S . Hence Cn A S ⊆ Th N S . (Anything true in every model of A S is true in this model.) What is not so obvious is that equality holds. We will prove this by considering arbitrary models of A S . What can be said of an arbitrary model A = (|A|; 0A , SA ) of the axioms A S ? SA must be a one-to-one map of |A| onto |A| − {0A }, by S1, S2, and S3. And by S4.n, there can be no loops of size n. Thus |A| must contain the “standard” points: 0A → SA (0A ) → SA (SA (0A )) → . . . , which are all distinct. The arrow here indicates the action of SA . There may or may not be other points. If there is another point a in |A|, then there will be the successor of a, its successor, etc. Not only that, but since (by S3) each nonzero element has a predecessor (something of which it is the successor) which is (by S2) unique, |A| must contain the predecessor of a, its predecessor, etc. These must all be distinct lest

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there be a ﬁnite loop. Thus a belongs to a “Z-chain”: · · · ∗ → ∗ → a → SA (a) → SA (SA (a)) → · · · . (We refer to these as Z-chains because they are arranged like the set Z of all integers {. . . , −1, 0, 1, 2, . . .}.) There can be any number of Z-chains. But any two Z-chains must be disjoint, as S2 prohibits merging. Similarly, any Z-chain must be disjoint from the standard part. This can be restated in another way. Say that two points a and b in |A| are equivalent if the function SA can be applied a ﬁnite number of times to one point to yield the other point. This is an equivalence relation. (It is clearly reﬂexive and symmetric; the transitivity follows from the fact that SA is one-to-one.) The standard part of |A| is the equivalence class containing 0A . For any other point (if any) a in |A|, the equivalence class of a is the set generated from {a} by SA and its inverse. This equivalence class is the Z-chain described above. Conversely, any structure B (for this language) that has a standard part 0B → SB (0B ) → SB (SB (0B )) → · · · and a nonstandard part consisting of any number of separate Z-chains is a model of A S . (Check through the list of axioms in A S , and note that each is true in B.) We thus have a complete characterization of what the models of A S must look like. If a model A of A S has only countably many Z-chains, then |A| is countable. In general, if the set of Z-chains has cardinality1 λ, then altogether the number of points in |A| is ℵ0 + ℵ0 ·λ. By facts of cardinal arithmetic (cf. Chapter 0) this number is the larger of ℵ0 and λ. Hence ℵ0 if A has countably many Z-chains, card |A| = λ if A has an uncountable number λ of Z-chains. LEMMA 31A If A and A are models of A S having the same number of Z-chains, then they are isomorphic. PROOF. There is a unique isomorphism between the standard part of A and the standard part of A . By hypothesis we are given a one-to-one correspondence between the set of Z-chains of A and the set of Z-chains of A ; thus each chain of A is paired with a chain of A . Clearly any two Z-chains are isomorphic. By combining all the individual isomorphisms (which uses the axiom of choice) we have an isomorphism of A onto A . Thus a model of A S is determined to within isomorphism by its number of Z-chains. For N S this number is zero, but any number is possible.

1

To avoid uncountable cardinals, see Exercise 3.

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A Mathematical Introduction to Logic The reader should note that there is no sentence of the language which says, “There are no Z-chains.” In fact, there is no set of sentences such that a model A of A S satisﬁes iff A has no Z-chains. For by the LST theorem there is an uncountable structure A with A ≡ N S . But A has uncountably many Z-chains and N S has none. THEOREM 31B Let A and B be uncountable models of A S of the same cardinality. Then A is isomorphic to B. PROOF. By the above discussion, A has card A Z-chains, and B has card B Z-chains. Since card A = card B, they have the same number of Z-chains and hence are isomorphic. THEOREM 31C Cn A S is a complete theory. PROOF. Apply the Lo´s–Vaught test of Section 2.6. The preceding theorem asserts that the theory Cn A S is categorical in any uncountable power. Furthermore, A S has no ﬁnite models. Hence the Lo´s–Vaught test applies. COROLLARY 31D

Cn A S = Th N S .

PROOF. We have Cn A S ⊆ Th N S ; the ﬁrst theory is complete and the second is satisﬁable.

COROLLARY 31E

Th N S is decidable.

PROOF. Any complete and axiomatizable theory is decidable (by Corollary 25G). A S is a decidable set of axioms for this theory.

Elimination of Quantiﬁers Once one knows a theory to be decidable, it is tempting to try to ﬁnd a realistically practical decision procedure. We will give such a procedure for Th N S , based on “elimination of quantiﬁers.” DEFINITION. A theory T admits elimination of quantiﬁers iff for every formula ϕ there is a quantiﬁer-free formula ψ such that T |= (ϕ ↔ ψ). Actually it is enough to consider only formulas ψ of a rather special form: THEOREM 31F Assume that for every formula ϕ of the form ∃ x(α0 ∧ · · · ∧ αn ), where each αi is an atomic formula or the negation of an atomic formula, there is a quantiﬁer-free formula ψ such that T |= (ϕ ↔ ψ). Then T admits elimination of quantiﬁers.

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PROOF. First we claim that we can ﬁnd a quantiﬁer-free equivalent for any formula of the form ∃ x θ for quantiﬁer-free θ . We begin by putting θ into disjunctive normal form (Corollary 15C). The resulting formula, ∃ x[(α0 ∧ · · · ∧ αm ) ∨ (β0 ∧ · · · ∧ βn ) ∨ · · · ∨ (ξ0 ∧ · · · ∧ ξt )], is logically equivalent to ∃ x(α0 ∧ · · · ∧ αm ) ∨ ∃ x(β0 ∧ · · · ∧ βn ) ∨ · · · ∨ ∃ x(ξ0 ∧ · · · ∧ ξt ). By assumption, each disjunct of this formula can be replaced by a quantiﬁer-free formula. We leave it to the reader to show (in Exercise 2) that by using the above paragraph one can obtain a quantiﬁer-free equivalent for an arbitrary formula. In the special case where the theory in question is the theory Th A of a structure A, the deﬁnition can be restated: Th A admits elimination of quantiﬁers iff for every formula ϕ there is a quantiﬁer-free formula ψ such that ϕ and ψ are “equivalent in A”; i.e., |=A (ϕ ↔ ψ)[s] for any map s of the variables into |A|. THEOREM 31G PROOF.

Th N S admits elimination of quantiﬁers.

By the preceding theorem, it sufﬁces to consider a formula ∃ x(α0 ∧ · · · ∧ αq ),

where each αi is atomic or is the negation of an atomic formula. We will describe how to replace this formula by another that is quantiﬁer-free. The equivalence of the new formula to the given one will, in fact, be a consequence of A S ; see Exercise 3. In the language of N S the only terms are of the form Sk u, where u is 0 or a variable. The only atomic formulas are equations. We may suppose that the variable x occurs in each αi . For if x does not occur in α, then ∃ x(α ∧ β) |==| α ∧ ∃ x β. Thus each αi has the form Sm x = Sn u or the negation of this equation, where u is 0 or a variable. We may further suppose u is different from x, since Sm x = Sn x could be replaced by 0 = 0 if m = n, and by 0 = 0 if m =

n.

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A Mathematical Introduction to Logic Case 1: Each αi is the negation of an equation. Then the formula may be replaced by 0 = 0. (Why?) Case 2: There is at least one αi not negated; say α0 is Sm x = t, where the term t does not contain x. Since the solution for x must be non-negative, we replace α0 by t= 0 ∧ ··· ∧ t = Sm−1 0 (or by 0 = 0 if m = 0). Then in each other α j we replace, say, Sk x = u ﬁrst by Sk+m x = Sm u, which in turn becomes Sk t = Sm u. We now have a formula in which x no longer occurs, so the quantiﬁer may be omitted. There are several interesting by-products of the quantiﬁer elimination procedure. For one, we get an alternative proof of the completeness of Cn A S . Suppose we begin with a sentence σ . The quantiﬁer elimination procedure gives a quantiﬁer-free sentence τ such that (by Exercise 3) A S |= (σ ↔ τ ). Now we claim that either A S |= τ or A S |= ¬ τ . For τ is built up from atomic sentences by means of ¬ and →. An atomic sentence must be of the form Sk 0 = Sl 0 and is deducible from A S if k = l, but is refutable (i.e., its negation is deducible)

l. (In fact, just {S1, S2} sufﬁces for this.) Since every from A S if k = atomic sentence can be deduced or refuted, so can every quantiﬁerfree sentence. This establishes the claim. And so either A S |= σ or A S |= ¬ σ . Another by-product concerns the problem of deﬁnability in N S ; see Exercises 4 and 5. For any formula ϕ in which just v1 and v2 occur free we now can ﬁnd a quantiﬁer-free ψ (with the same variables free) such that Th N S |= ∀ v1 ∀ v2 (ϕ ↔ ψ); i.e., |=NS ∀ v1 ∀ v2 (ϕ ↔ ψ). Thus the relation ϕ deﬁned is also deﬁnable by a quantiﬁer-free formula.

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Exercises 1. Let A∗S be the set of sentences consisting of S1, S2, and all sentences of the form ϕ(0) → ∀ v1 (ϕ(v1 ) → ϕ(Sv1 )) → ∀ v1 ϕ(v1 ), where ϕ is a wff (in the language of N S ) in which no variable except v1 occurs free. Show that A S ⊆ Cn A∗S . Conclude that Cn A∗S = Th N S . (Here ϕ(t) is by deﬁnition ϕtv1 . The sentence displayed above is called the induction axiom for ϕ.) 2. Complete the proof of Theorem 31F. Suggestion: Use induction. 3. The proof of quantiﬁer elimination for Th N S showed how, given a formula ϕ, to ﬁnd a quantiﬁer-free ψ. Show that A S |= (ϕ ↔ ψ) without using the completeness of Cn A S . (This yields an alternative proof of the completeness of Cn A S , not involving Z-chains or the Lo´s–Vaught test.) 4. Show that a subset of N is deﬁnable in N S iff either it is ﬁnite or its complement (in N) is ﬁnite. 5. Show that the ordering relation { m, n | m < n in N} is not deﬁnable in N S . Suggestion: It sufﬁces to show there is no quantiﬁer-free deﬁnition of ordering. Call a relation R ⊆ N × N linear if it can be covered by a ﬁnite number of lines. Call R colinear if it is the complement of a linear relation. Show that any relation deﬁnable in N S is either linear or colinear. And that the ordering relation is neither linear nor colinear. 6. Show that Th N S is not ﬁnitely axiomatizable. Suggestion: Show that no ﬁnite subset of A S sufﬁces, and then apply Section 2.6.

SECTION 3.2 Other Reducts of Number Theory1 First let us add the ordering symbol < to the language. The intended structure is N L = (N; 0, S, <).

1

This section may be omitted without disastrous effects.

194

A Mathematical Introduction to Logic We want to show that the theory of this structure is (like Th N S ) decidable and also admits elimination of quantiﬁers. But unlike Th N S , it is ﬁnitely axiomatizable and is not categorical in any inﬁnite cardinality. As axioms of Th N L we will take the ﬁnite set A L consisting of the six sentences listed below. Here x ≤ y is, of course, an abbreviation for (x < y ∨ x = y), and x ≤ y abbreviates the negation of this formula. ∀y

(y = 0 → ∃ x y = Sx)

(S3)

∀x ∀y

(x < Sy ↔ x ≤ y)

(L1)

∀x

x< 0

(L2)

∀x ∀y

(x < y ∨ x = y ∨ y < x)

(L3)

∀x ∀y

(x < y → y < x)

(L4)

∀x ∀y∀z

(x < y → y < z → x < z)

(L5)

On the one hand, it is easy to see that all six axioms are true in N L . Thus Cn A L ⊆ Th N L . On the other hand, the opposite inclusion is not obvious, and requires proof. We begin by listing some consequence of these axioms. (1)

A L ! ∀ x x < Sx.

PROOF. (2)

AL ! ∀ x x < x.

PROOF. (3)

(trichotomy).

For “→” use L3. For “←” use L4 and (2).

A L ! ∀ x ∀ y(x < y ↔ Sx < Sy).

PROOF.

From A L we can deduce the biconditionals: x < y ↔ y ≤ x ↔y< Sx ↔ Sx ≤ y ↔ Sx < Sy

(5)

In L4 take y to be x.

A L ! ∀ x ∀ y(x < y ↔ y ≤ x)

PROOF. (4)

In L1 take y to be x.

by (3); by L1; by (3); by L1.

A L ! S1 and A L ! S2.

PROOF. S1 follows from L2 and (1). S2 comes from (4) by use of L3 and (2). (6)

A L ! S4.n for n = 1, 2, . . . .

PROOF.

This follows from (1) and (2), by using L5.

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Thus any model A of A L is (when we ignore

The theory Cn A L admits elimination of quantiﬁers.

Again we consider a formula ∃ x(β0 ∧ · · · ∧ β p ),

where each βi is atomic or the negation of an atomic formula. The terms are, as in Section 3.1, of the form Sk u, where u is 0 or a variable. There are two possibilities for atomic formulas, Sk u = Sl t

and

Sk u < Sl t.

t2 by 1. We can eliminate the negation symbol. Replace t1 < t2 < t1 ∨ t1 = t2 and replace t1 = t2 by t1 < t2 ∨ t2 < t1 . (This is justiﬁed by L3 and L4.) By regrouping the atomic formulas and noting that ∃ x(ϕ ∧ ψ) |==| ∃ x ϕ ∧ ∃ x ψ, we may again reach formulas of the form ∃ x(α0 ∧ · · · ∧ αq ), where now each αi is atomic. 2. We may suppose that the variable x occurs in each αi . This is because if x does not occur in α, then ∃ x(α ∧ β) |==| α ∧ ∃ x β. Furthermore, we may suppose that x occurs on only one side of the equality or inequality αi . For Sk x = Sl x can be dealt with as in Section 3.1. Sk x < Sl x can be replaced by 0 = 0 if k < l, and 0= 0 otherwise. (This is justiﬁed by L1 and L4.) Case 1: Suppose that some αi is an equality. Then we can proceed as in case 2 of the quantiﬁer-elimination proof of Theorem 31G. Case 2: Otherwise each αi is an inequality. Then the formula can be rewritten ti < Sm i x ∧ Sn j x < u j . ∃x

i

j

indexed by i, so (Here i indicates the conjunction of formulas γ0 ∧ γ1 ∧· · · ∧ γk can be abbreviated i γi .) In the ﬁrst coni junction, i ti < Sm x, we have the lower bounds on x; in the second conjunction, j Sn j x < u j , we have the upper bounds. If the second conjunction is empty (i.e., if there are no upper bounds

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A Mathematical Introduction to Logic on x), then we can replace the formula by 0 = 0. (Why?) If the ﬁrst conjunction is empty (i.e., if there are no lower bounds on x), then we can replace the formula by Sn j 0 < u j , j

which asserts that zero satisﬁes the upper bounds. Otherwise, we rewrite the formula successively as ∃x (ti < Sm i x ∧ Sn j x < u j ). (1) i, j

∃x

(Sn j ti < Sm i +n j x < Sm i u j ).

i, j

⎛

⎝

⎞

Sn j +1 ti < Sm i u j ⎠ ∧

i, j

Sn j 0 < u j .

(2)

(3)

j

This last formula says “any lower bound plus one satisﬁes any upper bound, and furthermore zero satisﬁes any upper bound.” This implies that there is a gap between the greatest lower bound and the least upper bound, whence there is a solution for x. The second part guarantees that the solution for x is not forced to be negative. In each case, we have arrived at a quantiﬁer-free version of the given formula. COROLLARY 32B (a) Cn A L is complete. (b) Cn A L = Th N L .

(c) Th N L is decidable. PROOF. (a) The argument that followed the proof of Theorem 31G is applicable here also. (b) This follows from (a), since Cn A L ⊆ Th N L and Th N L is satisﬁable. For (c), we can use the fact that any complete axiomatizable theory is decidable. But the quantiﬁer elimination proof yields a more efﬁcient decision procedure. COROLLARY 32C A subset of N is deﬁnable in N L iff it is either ﬁnite or has ﬁnite complement. PROOF.

Compare Exercise 4 of the preceding section.

On the other hand, N L has more deﬁnable binary relations than has N S . For the ordering relation { m, n | m < n} is not deﬁnable in N S , by Exercise 5 of the preceding section. COROLLARY 32D

The addition relation, { m, n, p | m + n = p},

is not deﬁnable in N L .

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PROOF. If we could deﬁne addition, we could then deﬁne the set of even natural numbers. But this set is neither ﬁnite nor has ﬁnite complement. Now suppose we augment the language by the addition symbol +. The intended structure is N A = (N; 0, S, <, +). The theory of this structure is also decidable, as we will prove shortly. But to keep matters from getting even more complicated, we will avoid listing any convenient set of axioms for the theory. The nonstandard models of Th N A must also be models of Th N L . So they have a standard part, followed by some Z-chains. But ordering among the Z-chains can no longer be arbitrary. Let A be a nonstandard model of Th N A . The ordering

THEOREM 32E (PRESBURGER, 1929) The theory of the structure N A = (N; 0, S, <, +) is decidable. The proof is again based on a quantiﬁer elimination procedure. The theory of N A itself does not admit elimination of quantiﬁers. For example, the formula deﬁning the set of even numbers ∃ y v1 = y + y is not equivalent to any quantiﬁer-free formula. We can overcome this by adding a new symbol ≡2 for congruence modulo 2. Similarly, we add symbols ≡3 , ≡4 , . . . . The intended structure for this expanded language is N≡ = (N; 0, S, <, +, ≡2 , ≡3 , . . .), where ≡k is the binary relation of congruence modulo k. It turns out that the theory of this structure does admit elimination of quantiﬁers. This by itself does not imply that the theory of either structure is decidable. After all, we can start with any structure, and expand

198

A Mathematical Introduction to Logic it to a structure having additional relations until a structure is obtained that admits elimination of quantiﬁers. To obtain decidability, we must show that we can, given a sentence σ , (1) effectively ﬁnd a quantiﬁer-free equivalent σ , and then (2) effectively decide if σ is true. We will now give the quantiﬁer elimination procedure for Th N≡ . For a term t and a natural number n, let nt be the term t + t + · · · + t, with n summands. 0t is 0. Then any term can be expanded to one of the form Sn 0 0 + n 1 x 1 + · · · + n k x k for k ≥ 0, n i ≥ 0 (where the xi ’s are variables). For example, S(x + S0) + Sy becomes S3 0 + x + y. As usual we begin with a formula ∃ y(β1 ∧ · · · ∧ βn ), where βi is an atomic formula or the negation of one. 1. Eliminate negation. Replace ¬(t1 = t2 ) by (t1

= ≡m < <

u, u, u, ny + t,

where u and t are terms not containing y. In what follows we will take the liberty of writing these formulas with a subtraction symbol: ny ny ny u−t

= ≡m < <

u − t, u − t, u − t, ny.

These are merely abbreviations for the formulas without subtraction obtained by transposing terms.

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For example, we might have at this point the formula ∃ y(w < 4y ∧ 2y < u ∧ 3y < v ∧ y ≡3 t), where t, u, v, and w are terms not containing y. 2. Uniformize the coefﬁcients of y. Let p be the least common multiple of the coefﬁcients of y. Each atomic formula can be converted to one in which the coefﬁcient of y is p, by “multiplying through” by the appropriate factor. This is obviously legitimate for equalities and inequalities. In the case of congruences one must remember to raise the modulus also: a ≡m b

iff

ka ≡km kb.

In the example above p is 12, and we obtain ∃ y(3w < 12y ∧ 12y < 6u ∧ 12y < 4v ∧ 12y ≡36 12t). 3. Eliminate the coefﬁcient of y. Replace py by x and add the new conjunct x ≡ p 0. (In place of ∃ y · · · 12y · · · we can equally well have, “There exists a multiple x of 12 such that · · · x · · · .”) Our example is now converted to ∃ x(3w < x ∧ x < 6u ∧ x < 4v ∧ x ≡36 12t ∧ x ≡12 0). 4. Special case. If one of the atomic formulas is an equality, x + t = u, then we can replace ∃x θ by x θu−t ∧ t ≤ u.

Here replacement of x by “u −t” is the natural thing; we transpose terms to compensate for the absence of subtraction. For example, x (x ≡m v)u−t

is

u ≡m v + t.

5. We may assume henceforth that = does not occur. So we have a formula of the form ∃ x[r0 − s0 < x ∧ · · · ∧ rl−1 − sl−1 < x ∧ x < t0 − u 0 ∧ · · · ∧ x < tk−1 − u k−1 ∧ x ≡m 0 v0 − w0 ∧ · · · ∧ x ≡m n−1 vn−1 − wn−1 ], where ri , si , ti , u i , vi , and wi are terms not containing x. This can be abbreviated ⎡ ⎤ ∃ x ⎣ rj − sj < x ∧ x < ti − u i ∧ x ≡m i vi − wi ⎦. j

i

i

If there are no congruences (i.e., n = 0), then the formula asserts that there is a nonnegative space between the lower and

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A Mathematical Introduction to Logic upper bounds. We can replace the formula by the quantiﬁer-free formula: (r j − s j ) + S0 < ti − u i ∧ 0 < ti − u i . i

i

Let M be the least common multiple of the moduli m 0 , . . . , m n−1 . Then a + M ≡m i a. So as a increases, the pattern of residues of a modulo m 0 , . . . , m n−1 has period M. Thus, in searching for a solution to the congruences, we need only search M consecutive integers. We now have a formula that asserts the existence of a natural number which is not less than certain lower bounds L 1 , . . . , L l and which satisﬁes certain upper bounds and certain congruences. If there is such a solution, then one of the following is a solution: L 1 , L 1 + 1, . . . , L 1 + M − 1, L 2 , L 2 + 1, . . . , L 2 + M − 1, ··· L l , L l + 1, . . . , L l + M − 1, 0, 1, . . . , M − 1. (The last line is needed to cover the case in which every L j is negative. To avoid having to treat this line as a special case, we will add a new lower bound of 0. That is, let rl = 0 and sl = S0 so that rl − sl < x is a formula 0 < x + S0 asserting that x is nonnegative. We now have l + 1 lower bounds.) Our formula (asserting the existence of a solution for x) can now be replaced by a quantiﬁer-free disjunction that asserts that one of the numbers in the above matrix is a nonnegative solution: ri − si < (r j − s j ) + Sq 0 j≤l 1≤q≤M i≤l

∧

(r j − s j ) + Sq 0 < ti − u i

i

∧

(r j − s j )+S 0 ≡m i vi − wi . q

i

In our continuing example we have, after adding the new lower bound on x, ∃ x(3w < x ∧ 0 < x + S0 ∧ x < 6u ∧ x < 4v ∧ x ≡36 12t ∧ x ≡12 0). The quantiﬁer-free equivalent is a disjunction of 72 conjunctions. Each conjunction has six constituents.

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This proves half of the theorem. If we are given a sentence σ , the above procedure tells us how to ﬁnd effectively a quantiﬁerfree sentence τ (in the language of N≡ ) that is true (in the intended structure) iff σ is. Now we must decide if τ is true. But this is easy. It is enough to look at atomic sentences. Any variable-free term can be put in the form Sn 0. Then, for example, Sn 0 ≡m S p 0

is true iff n ≡m p.

Thus we have a decision procedure for Th N A . In 1974 Michael Fischer and Michael Rabin showed, however, that there is no decision procedure that is fast enough to be feasible for very long formulas. A set D of natural numbers is said to be periodic if for some positive p, any number n is in D iff n + p is in D. D is eventually periodic iff there exist positive numbers M and p such that for all n greater than M, n ∈ D iff n + p ∈ D. THEOREM 32F A set of natural numbers is deﬁnable in (N; 0, S, <, +) iff it is eventually periodic. PROOF. Exercise 1 asserts that every eventually periodic set is deﬁnable. Conversely, suppose D is deﬁnable. Then D is deﬁnable in N≡ by a quantiﬁer-free formula (whose only variable is v1 ). Since the class of eventually periodic sets is closed under union, intersection, and complementation, it sufﬁces to show that every atomic formula in the language of N≡ whose only variable is v1 deﬁnes an eventually periodic set. There are only four possibilities: nv1 + t nv1 + t u nv1 + t

= < < ≡m

u, u, nv1 + t, u,

where u and t are numerals. The ﬁrst two formulas deﬁne ﬁnite sets (which eventually have period 1), the third deﬁnes a set with ﬁnite complement, and the last formula deﬁnes a periodic set with period m. COROLLARY 32G

The multiplication relation { m, n, p | p = m · n in N}

is not deﬁnable in (N; 0, S, <, +). PROOF. If we had a deﬁnition of multiplication, we could then use that to deﬁne the set of squares. But the set of squares is not eventually periodic.

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A Mathematical Introduction to Logic

Exercises 1. Show that any eventually periodic set of natural numbers is deﬁnable in the structure N A . 2. Show that in the structure (N; +) the following relations are deﬁnable: (a) Ordering, { m, n | m < n}. (b) Zero, {0}. (c) Successor, { m, n | n = S(m)}. 3. Let A be a model of Th N L (or equivalently a model of A L ). For a and b in |A| deﬁne the equivalence relation: a ∼ b ⇐⇒ SA can be applied a ﬁnite number of times to one of a, b to reach the other. Let [a] be the equivalence class to which a belongs. Order equivalence classes by [a] ≺ [b]

iff a

and a ∼ b.

Show that this is a well-deﬁned ordering on the set of equivalence classes. 4. Show that the theory of the real numbers with its usual ordering, Th(R; <), admits elimination of quantiﬁers. (Assume that the language includes equality.)

SECTION 3.3 A Subtheory of Number Theory We now return to the full language of number theory, as described in Section 3.0. The parameters of the language are ∀, 0, S, <, +, ·, and E. The intended structure for this language is N = (N; 0, S, <, +, ·, E). Actually in (N; ·, E) we can deﬁne {0}, S, <, and +. (See Exercise 1.) As we will show in Section 3.8, in (N; +, ·) we can deﬁne E as well as 0, S, and <. So there are ways in which we could economize. The luxury of having all these parameters (particularly E) will simplify some of the proofs. As we shall see, Th N is a very strong theory and is neither decidable nor axiomatizable. In order to prove this fact (and a number of related results), it will be strategically wise to select for study a ﬁnitely axiomatizable subtheory of Th N. As hinted at in Section 3.0, this subtheory should be strong enough to represent (in a sense to be made precise) facts about decidable sets. The subtheory we have selected is Cn A E ,

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where A E is the set consisting of the eleven sentences listed below. (As in the preceding section, x ≤ y abbreviates x < y ∨ x = y.)

Set AE of Axioms ∀x

Sx = 0

(S1)

∀x ∀y

(Sx = Sy → x = y)

(S2)

∀x ∀y

(x < Sy ↔ x ≤ y)

(L1)

∀x

x< 0

(L2)

∀x ∀y

(x < y ∨ x = y ∨ y < x)

(L3)

∀x

x + 0=x

(A1)

∀x ∀y

x + Sy = S(x + y)

(A2)

∀x

x · 0=0

(M1)

∀x ∀y

x · Sy = x · y + x

(M2)

∀x

xE0 = S0

(E1)

∀x ∀y

x E Sy = x E y · x

(E2)

Since N is a model of A E , we have Cn A E ⊆ Th N. But (as we will prove in Section 3.5) equality does not hold here. In fact, it can be shown that A E S3, where S3 is the sentence ∀ y(y =

0 → ∃ x y = Sx). The ﬁrst ﬁve axioms give us some, but not all, of the axioms regarding S and < that were useful in the preceding sections. The other six axioms are the “recursion” equations for addition, multiplication, and exponentiation. We ﬁrst show that certain simple sentences in Th N are deducible from A E . LEMMA 33A (a) A E ! ∀ x x < 0. (b) For any natural number k, A E ! ∀ x(x < Sk+1 0 ↔ x = S0 0 ∨ · · · ∨ x = Sk 0). Notice that (a) can be thought of as the k = −1 case of (b), where the empty disjunction is ⊥. The lemma tells us that A E “knows” that the numbers less than 7, for example, are exactly 0, 1, 2, 3, 4, 5, 6. So in any model of A E , the standard points — the ones denoted by numerals S k 0 — are ordered in the natural way, and (by L3) the inﬁnite points, if any, are all larger than any standard point. PROOF. Part (a) is L2. For (b) we use induction (in English) on k. We have as a consequence of L1, x < S0 ↔ x < 0 ∨ x = 0,

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A Mathematical Introduction to Logic which together with L2 gives x < S0 ↔ x = 0, which is the k = 0 case of (b). For the inductive step we again apply L1: x < Sk+1 0 ↔ x < Sk 0 ∨ x = Sk 0. By the inductive hypothesis, x < Sk 0 can be replaced by x = S0 0 ∨ · · · ∨ x = Sk−1 0, whereby we obtain (b).

LEMMA 33B For any variable-free term t, there is a unique natural number n such that A E ! t = Sn 0. PROOF. The uniqueness is immediate. (Why? Because A E , weak as it is, at least knows, by S1, that 7 = 0, and by S2 80 times, that 87 = 80.) For the existence half, we use induction on t. If t is 0, we take n = 0. If t is Su, then by the inductive hypothesis A E ! u = Sm 0 for some m. Hence A E ! t = Sm+1 0. Now suppose t is u 1 + u 2 . By the inductive hypothesis A E ! t = Sm 0 + Sn 0 for some m and n. We now apply A2 n times and A1 once to obtain A E ! t = Sm+n 0. The arguments for multiplication and exponentiation are similar. As a special case of this lemma we have “2 + 2 = 4” (i.e., S2 0 + S 0 = S4 0) as a consequence of A E . A E is at least smart enough to evaluate variable-free terms. And the proof shows more than this. The proof provides exact instructions for how, given such a term t, to ﬁnd effectively the unique number n such that A E ! t = Sn 0. 2

THEOREM 33C For any quantiﬁer-free sentence τ true in N, A E ! τ . PROOF. Exercise 2. Start with the atomic sentences; these will be of the form t1 = t2 or t1 < t2 for variable-free terms t1 and t2 . Show that A E proves τ if τ is true in N, and refutes τ (i.e., proves ¬ τ ) if τ is false in N. Later on, we will improve on Theorem 33C by allowing τ to contain “bounded quantiﬁers”; see Theorem 33I. A simpliﬁed notation (used earlier in Section 2.7) for substitution will be helpful in the coming pages: ϕ(t) = ϕtv1 , v ϕ(t1 , t2 ) = ϕtv1 1 t22 , and so forth. Thus ϕ = ϕ(v1 ) = ϕ(v1 , v2 ). Usually the term substituted

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will be a numeral, for example

v ϕ(Sa 0, Sb 0) = ϕSva10 Sb20 .

But at times we will also substitute other terms, e.g., ϕ(x) = ϕxv1 , where x is a variable. If, however, x is not substitutable for v1 in ϕ, then we must take ϕ(x) = ψxv1 , where ψ is a suitable alphabetic variant of ϕ. In the next proof (and elsewhere in this chapter) we make use of the following consequence of the substitution lemma of Section 2.5: For a formula ϕ in which at most v1 , . . . , vn occur free and for natural numbers a1 , . . . , an , |=N ϕ[[a1 , . . . , an ]] ⇔ |=N ϕ(Sa1 0, . . . , San 0). An existential (∃1 ) formula is one of the form ∃ x1 · · · ∃ xk θ , where θ is quantiﬁer-free. The following result improves Theorem 33C: COROLLARY 33D AE ! τ .

If τ is an existential sentence true in N, then

PROOF. If ∃ v1 ∃ v2 θ is true in N, then for some natural numbers m and n, θ (Sm 0, Sn 0) is true in N. As this is a quantiﬁer-free true sentence, it is deducible from A E . But it in turn logically implies ∃ v1 ∃ v2 θ . On the other hand, it is known that there are true universal (∀1 ) sentences (i.e., of the form ∀ x1 · · · ∀ xk θ for quantiﬁer-free θ ) that are not in Cn A E .

Representable Relations Let R be an m-ary relation on N; i.e., R ⊆ Nm . We know that a formula ρ (in which only v1 , . . . , vm occur free) deﬁnes R in N iff for every a1 , . . . , am in N, a1 , . . . , am ∈ R ⇔ |=N ρ[[a1 , . . . , am ]] ⇔ |=N ρ(Sa1 0, . . . , Sam 0). (The last condition here is equivalent to the preceding one by the substitution lemma.) We can recast this into two implications: a1 , . . . , am ∈ R ⇒ |=N ρ(Sa1 0, . . . , Sam 0), / R ⇒ |=N ¬ ρ(Sa1 0, . . . , Sam 0). a1 , . . . , am ∈ We will say that ρ also represents R in the theory Cn A E if in these two implications the notion of truth in N can be replaced by the stronger notion of deducibility from A E . More generally, let T be any theory in a language with 0 and S. Then ρ represents R in T iff for every a1 , . . . , am in N: a1 , . . . , am ∈ R ⇒ ρ(Sa1 0, . . . , Sam 0) ∈ T, / R ⇒ ( ¬ ρ(Sa1 0, . . . , Sam 0)) ∈ T. a1 , . . . , am ∈

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A Mathematical Introduction to Logic For example, ρ represents R in the theory Th N iff ρ deﬁnes R in N. But ρ represents R in Cn A E iff for all a1 , . . . , am : a1 , . . . , am ∈ R ⇒ A E ! ρ(Sa1 0, . . . , Sam 0), / R ⇒ A E ! ¬ ρ(Sa1 0, . . . , Sam 0). a1 , . . . , am ∈ The equality relation on N, for example, is represented in Cn A E by the formula v1 = v2 . For m = n ⇒ ! Sm 0 = Sn 0, m = n ⇒ {S1, S2} ! ¬ Sm 0 = Sn 0. A relation is representable in T iff there exists some formula that represents it in T . The concept of representability should be compared with that of deﬁnability. In both cases we are somehow describing relations on the natural numbers by formulas. In the case of deﬁnability, we ask about the truth of sentences in the interpretation. In the case of representability in Cn A E , we ask instead about the deducibility of sentences from the axioms. Say that a formula ϕ, in which no variables other than v1 , . . . , vm occur free, is numeralwise determined by A E iff for every m-tuple a1 , . . . , am of natural numbers, either A E ! ϕ(Sa1 0, . . . , Sam 0) or A E ! ¬ ϕ(Sa1 0, . . . , Sam 0) THEOREM 33E A formula ρ represents a relation R in Cn A E iff (1) ρ is numeralwise determined by A E , and (2) ρ deﬁnes R in N. PROOF. We use the fact that N is a model of A E . If ρ represents R in Cn A E , then it is clear that (1) holds; (2) holds since “A E !” implies “|=N .” Conversely, if (1) and (2) hold, then we have a1 , . . . , am ∈ R ⇒ |=N ρ(Sa1 0, . . . , Sam 0) ⇒ A E ¬ ρ(Sa1 0, . . . , Sam 0) ⇒ A E ! ρ(Sa1 0, . . . , Sam 0) Similarly for the complement of R and ¬ ρ.

by (2) since N is a model of A E by (1).

Church’s Thesis We now turn to the relationship of the concepts of representability and decidability.

THEOREM 33F Assume that R is a relation representable in a consistent axiomatizable theory. Then R is decidable.

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PROOF. Say that ρ represents R in the consistent axiomatizable theory T . Recall that T is effectively enumerable (Corollary 25F). The decision procedure is as follows: Given a1 , . . . , am , enumerate the members of T . If, in the enumeration, ρ(Sa1 0, . . . , Sam 0) is found, then we are done and a1 , . . . , am ∈ R. If, in the enumeration, ¬ ρ(Sa1 0, . . . , Sam 0) is / R. found, then we are done and a1 , . . . , am ∈ By the representability, one sentence or the other always appears eventually, whereupon the procedure terminates. Since T is consistent, the answer given by the procedure is correct.

COROLLARY 33G Any relation representable in a consistent ﬁnitely axiomatizable theory is decidable.

What about the converse to the above corollary? We cannot really hope to prove the converse on the basis of our informal notion of decidability. For our informal approach is usable only for giving lower bounds on the class of decidable relations (i.e., for showing that certain relations are decidable) and is unsuited to giving upper bounds (i.e., for showing undecidability). It is nevertheless possible to make plausibility arguments in support of the converse. This will be easier to do at the end of Section 3.4 than here. Roughly, the idea is that in a ﬁnite number of axioms we could capture the (ﬁnitely long) instructions for the decision procedure. The assertion that both the above corollary and its converse are correct is generally known as Church’s thesis. This assertion is not really a mathematical statement susceptible to proof or disproof; rather it is a judgment that the correct formalization of the informal notion of decidability is by means of representability in consistent and ﬁnitely axiomatizable theories. DEFINITION. A relation R on the natural numbers is recursive iff it is representable in some consistent ﬁnitely axiomatizable theory (in a language with 0 and S). Church’s thesis now can be put more succinctly: A relation is decidable iff it is recursive. Or perhaps more accurately: The concept of recursiveness is the correct precise counterpart to the informal concept of decidability. The situation is analogous to one encountered in calculus. An intuitively continuous function (deﬁned on an interval) is one whose graph you can draw without lifting your pencil off the paper. But to prove theorems, some formal counterpart of this notion is needed. And so one gives the usual deﬁnition of ε-δ-continuity. One should ask if the precise notion of ε-δ-continuity is an accurate formalization of intuitive continuity. If anything, the class of ε-δ-continuous functions is too broad. It includes nowhere differentiable functions, whose graphs cannot be drawn without lifting the pencil. But accurate or not, the class

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A Mathematical Introduction to Logic of ε-δ-continuous functions has been found to be a natural and important class in mathematical analysis. Very much the same situation occurs with recursiveness. One should ask if the precise notion of recursiveness is an accurate formalization of the informal notion of decidability. Again, the precisely deﬁned class (of recursive relations) appears to be, if anything, too broad. It includes relations for which any decision procedure will, for large inputs, require so much computing time and memory (“scratchpad”) space as to make implementation absurd. Recursiveness corresponds to decidability in an idealized world, where length of computation and amount of memory are disregarded. But in any case, the class of recursive relations has been found to be a natural and important class in mathematical logic. Empirical evidence that the class of recursive relations is not too narrow is provided by the following: 1. Any relation considered thus far that mathematicians have felt was decidable has been found to be recursive. 2. Several people have tried giving precise deﬁnitions of idealized computing agents. The best-known such idealized agents are the “Turing machines,” introduced by Alan Turing in 1936. (A variation on that idea leads to the register machines described in Section 3.6.) The idea was to devise something that could carry out any effective procedure. In all cases, the class of relations having decision procedures executable by such a computing agent has been exactly the class of recursive relations. (Because of the importance of Turing’s analysis of effective computability, Church’s thesis is often called the Church–Turing thesis.) The fact that so many different (yet equivalent) deﬁnitions for the class of recursive relations have been found is some indication of the naturalness and importance of the concept. In this book we will continue to exclude the informal notion of decidability from nonstarred theorems. But in the remainder of the exposition we will accept Church’s thesis. For example, we will speak of a set’s being undecidable when we have a theorem stating it to be nonrecursive. Obviously any relation representable in Cn A E is recursive. We will prove later that the converse also holds; if a relation is representable in any consistent ﬁnitely axiomatizable theory, then it is representable in the one theory we have selected for special study. (This was, of course, a motivating factor in our selection.) The use of the word “recursive” in this context is the result of historical accident — even of historical error. Recently several mathematicians have argued that the word “computable” would more accurately reﬂect the intended ideas. But in the present context, we want to reserve the word “computable” for an informal concept, to be deﬁned next. For relations we have the informal concept of decidability; for functions the

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analogous concept is computability. (As notational shorthand, a string a1 , . . . , ak can be written as a .)

DEFINITION. A function f : Nk → N is computable iff there is an effective procedure that, given any k-tuple a of natural numbers, will produce f ( a ).

For example, addition and multiplication are computable. Effective procedures, using base-10 notation, for these functions are taught in the elementary schools. (Strictly speaking, in the concept of computability one should refer to being given numerals, not numbers. For it is numerals — strings of symbols like the triple 317 or the triple XCI — that can be communicated. Nonetheless, we will suppress this point.) On the other hand, of the uncountably many functions from Nk into N, only countably many can be computable, because there are only countably many effective procedures. We want to give a mathematical counterpart to the informal concept of computability, just as in the case of decidable relations. The clue to the correct counterpart is provided by the next theorem. Recall that any function f : Nk → N is also a (k + 1)-ary relation on N: a1 , . . . , ak , b ∈ f

⇐⇒

f (a1 , . . . , ak ) = b.

At one time it was popular to distinguish between the function and the relation (which was called the graph of the function). Current settheoretic usage takes a function to be the same thing as its graph. But we still have the two ways of looking at the function.

THEOREM 33H The following three conditions on a function f : Nk → N are equivalent: (a) f is computable. (b) When viewed as a relation, f is a decidable relation. (c) When viewed as a relation, f is an effectively enumerable relation.

PROOF. (a) ⇒ (b): Assume that f is computable; we will describe the decision procedure. Given a1 , . . . , ak , b , ﬁrst compute f (a1 , . . . , ak ). Then look to see if the result is equal to b. If it is say “yes,” otherwise say “no.” (b) ⇒ (c): Any decidable relation is effectively enumerable. For we can enumerate the set of all (k + 1)-tuples of numbers, and place on the output list those which meet the test of belonging to the relation. (c) ⇒ (a): Assume that we have an effective enumeration of (the graph of ) f . To compute f (a1 , . . . , ak ) we examine the (k + 1)-tuples in the enumeration until we ﬁnd the one that begins with a1 . . . , ak . Its last component is then the desired function value.

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A Mathematical Introduction to Logic Thus by using Church’s thesis, we can say that f is computable iff f (viewed as a relation) is recursive. The class of recursive functions is an interesting class even apart from its connection with incompleteness theorems of logic. It represents an upper bound to the class of functions that can actually be computed by programs for digital computers. The recursive functions are those which are calculable by digital computers, provided one ignores practical limitations on computing time and memory space. We can now describe our plans for this section and the next. Our basic goal is to obtain the theorems of Section 3.5. But some groundwork is required before we can prove those theorems; we must verify that a number of relations (intuitively decidable) and a number of functions (intuitively computable) are representable in Cn A E and hence are recursive. In the process we will show (Theorem 34A) that recursiveness is equivalent to representability in Cn A E . In the remainder of the present section we will establish general facts about representability, and will show, for example, that certain functions for encoding ﬁnite sequences of numbers into single numbers are representable. Then in Section 3.4 we apply these results to particular relations and functions related to the syntactical features of the formal language. The author is sufﬁciently realistic to know that many readers will be more interested in the theorems of Section 3.5 than in the preliminary spadework. If the reader is willing to believe that intuitively decidable relations are all representable in Cn A E , and intuitively computable functions are functionally representable (a concept we will deﬁne shortly) there, then most if not all of the proofs in this spadework become unnecessary. But it is hoped that the deﬁnitions and the statements of the results will still receive some attention.

Numeralwise Determined Formulas Theorem 33E tells us that we can show a relation to be representable in Cn A E by ﬁnding a formula that deﬁnes it in N and is numeralwise determined by A E . The next theorem will be useful in establishing numeralwise determination. THEOREM 33I (a) Any atomic formula is numeralwise determined by A E . (b) If ϕ and ψ are numeralwise determined by A E , then so are ¬ ϕ and ϕ → ψ. (c) If ϕ is numeralwise determined by A E , then so are the following formulas (obtained from ϕ by “bounded quantiﬁcation”): ∀ x(x < y → ϕ), ∃ x(x < y ∧ ϕ).

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PROOF. Part (a) follows from Theorem 33C. Part (b) is easy. It remains to prove part (c). We will consider a formula ∃ x(x < y ∧ ϕ(x, y, z)) in which just the variables y and z occur free. Consider two natural numbers a and b; we must show that either A E ! ∃ x(x < Sa 0 ∧ ϕ(x, Sa 0, Sb 0)) or A E ! ¬ ∃ x(x < Sa 0 ∧ ϕ(x, Sa 0, Sb 0)). Case 1: For some c less than a, A E ! ϕ(Sc 0, Sa 0, Sb 0).

(1)

(This case occurs iff ∃ x(x < Sa 0 ∧ ϕ(x, Sa 0, Sb 0)) is true in N.) We also have A E ! Sc 0 < Sa 0.

(2)

And the sentences in (1) and (2) logically imply the sentence ∃ x(x < Sa 0 ∧ ϕ(x, Sa 0, Sb 0)). Case 2: Otherwise for every c less than a, A E ! ¬ ϕ(Sc 0, Sa 0, Sb 0).

(3)

(This case occurs iff ∀ x(x < Sa 0 → ¬ ϕ(x, Sa 0, Sb 0)) is true in N.) We know from Lemma 33A that A E ! ∀ x(x < Sa 0 → x = S0 0 ∨ · · · ∨ x = Sa−1 0).

(4)

The sentence in (4) together with the sentences in (3) (for c = 0, . . . , a − 1) logically imply ∀ x(x < Sa 0 → ¬ ϕ(x, Sa 0, Sb 0)). And this is equivalent to ¬ ∃ x(x < Sa 0 ∧ ϕ(x, Sa 0, Sb 0)). This shows that ∃ x(x < y ∧ ϕ(x, y, z)) is numeralwise determined by A E . By applying this result to ¬ ϕ we obtain the fact that the dual formula, ∀ x(x < y → ϕ(x, y, z)), is numeralwise determined by A E as well. The argument in case 2 relied on the fact that the x quantiﬁer was bounded by Sa 0. We will see that it is possible for ¬ ψ(S0 0), ¬ ψ(S1 0), . . .

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A Mathematical Introduction to Logic all to be consequences of A E without having ∀ x ¬ ψ(x) be a consequence. The preceding theorem is a useful tool for showing many relations to be representable in Cn A E . For example, the set of primes is represented by S1 0 < v1 ∧ ∀ x(x < v1 → ∀ y(y < v1 → x · y = v1 )). This formula deﬁnes the primes in N, and by the preceding theorem is numeralwise determined by A E . It therefore represents the set of primes in Cn A E .

Representable Functions Often it is more convenient to work with functions than with relations. Let f : Nm → N be an m-place function on the natural numbers. A formula ϕ in which only v1 , . . . , vm+1 occur free will be said to functionally represent f (in the theory Cn A E ) iff for every a1 , . . . , am in N, ! A E ! ∀ vm+1 ϕ(Sa1 0, . . . , Sam 0, vm+1 ) ↔ vm+1 = S f (a1 ,...,am ) 0 . (Observe that the “←” half of this sentence is equivalent to ϕ(Sa1 0, . . . , Sam 0, S f (a1 ,...,am ) 0). The “→” half adds an assertion of uniqueness.) THEOREM 33J If ϕ functionally represents f in Cn A E , then it also represents f (as a relation) in Cn A E . PROOF, WITH m = 1. for any b:

Since ϕ functionally represents f , we have

A E ! ϕ(Sa 0, Sb 0) ↔ Sb 0 = S f (a) 0. If a, b ∈ f , i.e., if f (a) = b, then the right half of this biconditional is valid and we get A E ! ϕ(Sa 0, Sb 0). But otherwise the right half is refutable from A E (i.e., its negation is deducible), whence A E ! ¬ ϕ(Sa 0, Sb 0).

The converse of this theorem is false. But we can change the formula: THEOREM 33K Let f be a function on N that is (as a relation) representable in Cn A E . Then we can ﬁnd a formula ϕ that functionally represents f in Cn A E .

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PROOF. To simplify the notation we will take f to be a one-place function on N. The desired sentence, ∀ v2 [ϕ(Sa 0, v2 ) ↔ v2 = S f (a) 0], is equivalent to the conjunction of the two sentences and

ϕ(Sa 0, S f (a) 0)

(1)

∀ v2 [ϕ(Sa 0, v2 ) → v2 = S f (a) 0].

(2)

The sentence (1) is a theorem of A E whenever ϕ represents f . The sentence (2) is an assertion of uniqueness; we must construct ϕ in such a way that this will also be a theorem of A E . Begin with a formula θ known to represent f (as a binary relation). Let ϕ be θ (v1 , v2 ) ∧ ∀ z(z < v2 → ¬ θ(v1 , z)). We can then rewrite (2) as ∀ v2 [θ(Sa 0, v2 ) ∧ ∀ z(z < v2 → ¬ θ(Sa 0, z)) → v2 = S f (a) 0].

(2 )

To show this to be a theorem of A E it clearly sufﬁces to show that A E ∪ {θ (Sa 0, v2 ), ∀ z(z < v2 → ¬ θ(Sa 0, z))} ! v2 = S f (a) 0. Call this set of hypotheses (to the left of “!”) . Since L3 ∈ A E it sufﬁces to show that and

! v2 < S f (a) 0

(3)

v2 . ! S f (a) 0 <

(4)

It is easy to obtain (4), since from the last member of we get S f (a) 0 < v2 → ¬ θ(Sa 0, S f (a) 0) and we know that A E ! θ(Sa 0, S f (a) 0).

(5)

To obtain (3) we ﬁrst note that we have as theorems of A E , v2 < S f (a) 0 ↔ v2 = S0 0 ∨ · · · ∨ v2 = S f (a)−1 0

(6)

and ¬ θ (Sa 0, Sb 0)

for b = 0, . . . , f (a) − 1.

(7)

The formulas (6) and (7) imply the formula v2 < S f (a) 0 → ¬ θ(Sa 0, v2 ). Since θ (Sa 0, v2 ) ∈ , we have (3).

(8)

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A Mathematical Introduction to Logic This shows (2) to be a theorem of A E ; (5) and (8) show (1) to be a theorem of A E as well. We next want to show that certain basic functions are representable (in Cn A E ) and that the class of representable functions has certain closure properties. Henceforth in this section, when we say that a function or relation is representable, we will mean that it is representable in the theory Cn A E . But the phrase “in Cn A E ” will usually be omitted. In simple cases, an m-place function might be represented by an equation vm+1 = t. In fact, any such equation, when the variables in t are among v1 , . . . , vm , deﬁnes in N an m-place function f . (The value of f at a1 , . . . , am is the number assigned in N to t when vi is assigned ai , 1 ≤ i ≤ m.) Furthermore, we know that any equation is numeralwise determined by A E , so the equation represents f as a relation. In fact, it even functionally represents f , for the sentence ∀ vm+1 [vm+1 = t (Sa1 0, . . . , Sam 0) ↔ vm+1 = S f (a1 ,...,am ) 0] is logically equivalent to t (Sa1 0, . . . , Sam 0) = S f (a1 ,...,am ) 0, which is a quantiﬁer-free sentence true in N. (Here t (u 1 , . . . , u m ) is the term obtained by replacing v1 by u 1 , then v2 by u 2 , etc.) For example: 1. The successor function is represented (functionally) by the equation v2 = Sv1 . 2. Any constant function is representable. The m-place function that constantly assumes the value b is represented by the equation vm+1 = Sb 0. 3. The projection function (where 1 ≤ i ≤ m) Iim (a1 , . . . , am ) = ai is represented by the equation vm+1 = vi . 4. Addition, multiplication, and exponentiation are represented by the equations v3 = v1 + v2 , v3 = v1 · v2 , v3 = v1 E v2 , respectively.

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The reader should not be misled by these simple examples; not every representable function is representable by an equation. We next want to show that the family of representable functions is closed under composition. To simplify the notation, we will consider a one-place function f on N, where f (a) = g(h 1 (a), h 2 (a)). Suppose that g is functionally represented by ψ and h i by θi . To represent f it would be reasonable to try either ∀ y1 ∀ y2 (θ1 (v1 , y1 ) → θ2 (v1 , y2 ) → ψ(y1 , y2 , v2 )) or ∃ y1 ∃ y2 (θ1 (v1 , y1 ) ∧ θ2 (v1 , y2 ) ∧ ψ(y1 , y2 , v2 )). (Think of ψ(y1 , y2 , v2 ) as saying “g(y1 , y2 ) = v2 ” and think of θi (v1 , y1 ) as saying “h i (v1 ) = yi .” Then the ﬁrst formula translates, “For any y1 , y2 , if h 1 (v1 ) = y1 and h 2 (v1 ) = y2 , then g(y1 , y2 ) = v2 .” The second formula translates, “There exist y1 , y2 such that h 1 (v1 ) = y1 and h 2 (v1 ) = y2 and g(y1 , y2 ) = v2 .” Either one is a reasonable way of saying, “g(h 1 (v1 ), h 2 (v1 )) = v2 .” There are two choices, because when something is unique, either quantiﬁer can be used for it.) Actually either formula would work; let ϕ be ∀ y1 ∀ y2 (θ1 (v1 , y1 ) → θ2 (v1 , y2 ) → ψ(y1 , y2 , v2 )). Consider any natural number a; we have at our disposal ∀ v2 [ψ(Sh 1 (a) 0, Sh 2 (a) 0, v2 ) ↔ v2 = S f (a) 0].

(1)

∀ y1 [θ1 (Sa 0, y1 ) ↔ y1 = Sh 1 (a) 0].

(2)

∀ y2 [θ2 (Sa 0, y2 ) ↔ y2 = Sh 2 (a) 0].

(3)

And we want ∀ v2 (ϕ(Sa 0, v2 ) ↔ v2 = S f (a) 0),

(4)

i.e., ∀ v2 ( ∀ y1 ∀ y2 [θ1 (Sa 0, y1 ) → θ2 (Sa 0, y2 ) → ψ(y1 , y2 , v2 )] ↔ v2 = S f (a) 0).

(4)

But (1), (2), and (3) imply (4), as the reader is asked to verify in Exercise 4. More generally we have THEOREM 33L Let g be an n-place function, let h 1 , . . . , h n be mplace functions, and let f be deﬁned by f (a1 , . . . , am ) = g(h 1 (a1 , . . . , am ), . . . , h n (a1 , . . . , am )).

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A Mathematical Introduction to Logic From formulas functionally representing g and h 1 , . . . , h n we can ﬁnd a formula that functionally represents f . In the above proof we have m = 1 and n = 2. But the general case is proved in exactly the same way. In order to obtain a function such as f (a, b) = g(h(a), b), we note that

f (a, b) = g h I12 (a, b) , I22 (a, b) .

The above theorem then can be applied (twice) to show that f is representable (provided that g and h are). To facilitate discussion of functions with an arbitrary number of variables, we will use vector notation. For example, the equation in the above theorem can be written f ( a ) = g(h 1 ( a ), . . . , h n ( a )). Another important closure property of the functions representable in Cn A E is closure under the “least-zero” operator. THEOREM 33M Assume that the (m + 1)-place function g is representable and that for every a1 , . . . , am there is a b such that g(a1 , . . . , am , b) = 0. Then we can ﬁnd a formula that represents the m-place function f , where f (a1 , . . . , am ) = the least b such that g(a1 , . . . , am , b) = 0. (In vector notation we can rewrite this last equation: f ( a ) = the least b such that g( a , b) = 0. The traditional notation for the least-zero operator is f ( a ) = μb[g( a , b) = 0] and the operator is often called “the μ-operator.”) PROOF.

To simplify the notation we take m = 1; thus

f (a) = b

iff g(a, b) = 0

and for all

c < b, g(a, c) =

0.

If ψ represents g, then we can obtain a formula representing f (as a relation) simply by formalizing the right side of this equivalence: ψ(v1 , v2 , 0) ∧ ∀ y(y < v2 → ¬ ψ(v1 , y, 0)).

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This formula deﬁnes (the graph of ) f and is numeralwise deter mined by A E .

A Catalog We now construct a repertoire of representable (in Cn A E ) functions and relations, including in particular functions for encoding and decoding sequences. 0. As a consequence of Theorem 33I, any relation that has (in N) a quantiﬁer-free deﬁnition is representable. And the class of representable relations is closed under unions, intersections, and complements. And if R is representable, then so are { a1 , . . . , am , b | for all c < b, a1 , . . . , am , c ∈ R} and { a1 , . . . , am , b | for some c < b, a1 , . . . , am , c ∈ R}. For example, any ﬁnite relation has a quantiﬁer-free deﬁnition, as does the ordering relation. 1. A relation R is representable iff its characteristic function K R is. a ) = 1 when a ∈ R, and K R ( a) = 0 (K R is the function for which K R ( otherwise.) PROOF. (⇐) Say that R is a unary relation (a subset of N) and that K R is represented by ψ(v1 , v2 ). We claim that ψ(v1 , S0) represents R. For it deﬁnes R and is numeralwise determined by A E . (⇒) Say that ϕ(v1 ) represents R. Then (ϕ(v1 ) ∧ v2 = S0) ∨ ( ¬ ϕ(v1 ) ∧ v2 = 0) represents (the graph of ) K R , for the same reason as in the last paragraph. (Actually this formula even functionally represents K R , as the reader can verify.) 2. If R is a representable binary relation and f , g are representable functions, then { a | f ( a ), g( a ) ∈ R} is representable. Similarly for an m-ary relation R and functions f1, . . . , fm . PROOF. Its characteristic function at a has the value K R ( f ( a ), g( a )). Thus it is obtained from representable functions by composition. For example, suppose that R is a representable ternary relation. Then { x, y | y, x, x ∈ R}

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A Mathematical Introduction to Logic is representable, being # " $ x, y # I22 (x, y), I12 (x, y), I12 (x, y) ∈ R . In this way we can rearrange and repeat variables in describing a representable relation. 3. If R is a representable binary relation, then so is P = { a, b | for some c ≤ b, a, c ∈ R}. PROOF.

We have from catalog item 0 that if Q = { a, b | for some c < b, a, c ∈ R},

then Q is representable. And a, b ∈ P ⇔ a, S(b) ∈ Q

⇔ I12 (a, b), S I22 (a, b) ∈ Q. Hence by catalog item 2, P is representable.

More generally, if R is a representable (m + 1)-ary relation, then { a1 , . . . , am , b | for some c ≤ b, a1 , . . . , am , c ∈ R} is also representable. In vector notation this relation becomes { a , b | for some c ≤ b, a , c ∈ R}. Similarly { a , b | for all c ≤ b, a , c ∈ R} is representable. 4. The divisibility relation { a, b | a divides b in N} is representable. PROOF. We have a divides b iff for some q ≤ b, a ·q = b. We know that { a, b, q | a·q = b} is representable, as it has a quantiﬁer-free deﬁnition. Upon applying the above items, we get the divisibility relation. (In yet further detail, from catalog item 3 we get the representability of R = { a, b, c | for some q ≤ c, a · q = b} and a divides b iff a, b, b ∈ R.)

5. The set of primes is representable. 6. The set of pairs of adjacent primes is representable. PROOF. a, b is a pair of adjacent primes iff a is prime and b is prime and a < b and there does not exist any c < b such that

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a < c and c is prime. The right side of this equivalence is easily formalized by a numeralwise determined formula. Note (for future use in Section 3.8) that we have not yet used the fact that exponentiation is representable. Observe that as this catalog progresses, we are in effect building up a “language” L such that anything (any relation, any function) that is L-deﬁnable (in N) will be certain to be representable in our theory. Thus, Theorem 33I tells us that (a) atomic formulas are allowed in L, (b) all sentential connectives are permitted, and (c) bounded quantiﬁers can be used. (Unbounded quantiﬁers are not in general allowed.) Then our catalog gradually adds particular predicate symbols and function symbols; catalog item 6 adds a two-place predicate symbol for “adjacent primality”; and catalog item 7 will add a function symbol for the prime-listing function. Theorem 33L justiﬁes using these function symbols inside expressions of L. 7. The function whose value at a is pa , the (a + 1)st prime, is representable. (Thus p0 = 2, p1 = 3, p2 = 5, p3 = 7, p4 = 11, and so forth.) 2

PROOF. pa = b iff b is prime and there exists some c ≤ ba , such that (i)–(iii) hold: (i) 2 does not divide c. (ii) For any q < b and any r ≤ b, if q, r is a pair of adjacent primes, then for all j < c, q j divides c ⇐⇒ r j+1 divides c. (iii) ba divides c and ba+1 does not. This equivalence is not obvious, but at least the relation deﬁned by the right-hand side is representable. To verify the equivalence, ﬁrst note that if pa = b, then we can take c = 20 · 31 · 52 · . . . · paa . It is easy to check that this value of c meets all the conditions. Conversely, suppose c is a number meeting conditions (i)–(iii). We claim that c must be 20 · 31 · . . . · ba · powers of larger primes. Certainly the exponent of 2 in c is 0, by (i). We can use (ii) to work our way across to the prime b. But by (iii) the exponent of b is a, so b must be the (a + 1)st prime, pa .

220

A Mathematical Introduction to Logic This function will be very useful in encoding ﬁnite sequences of numbers into single numbers. Let

a0 , . . . , am = p0a0 +1 · · · · · pmam +1 % ai +1 = pi . i≤m

This holds also for m = −1; we deﬁne = 1. For example,

2, 1 = 23 · 32 = 72. The idea is that 72 safely encodes the pair 2, 1 . There are other ways to encode pairs of numbers and ﬁnite sequences of numbers. In Section 3.8, we will make use of a pairing function 1 [(a + b)2 + 3a + b] 2 that has the advantage of growing at a polynomial rate, unlike the growth rate of 2a+1 3b+1 . Here is a very different way to encode, for example, the numbers 24, 117, 11 (in that order). First we convert to numerals in base 9: 26, 140, 12. Secondly, we concatenate these numerals, separated by 9’s: 269140912. The triple is encoded by the number thereby designated (in base 10), that is, 269,140,912. This method may seem tricky, but it produces a result that is much smaller than 225 3118 512 , which requires 73 digits in base 10. J (a, b) =

8. For each m, the function whose value at a0 , . . . , am is a0 , . . . , am is representable. 9. There is a representable function (whose value at a, b is written (a)b ) such that for b ≤ m, ( a0 , . . . , am )b = ab . (This is our “decoding” function. For example, (72)0 = 2 and (72)1 = 1.) PROOF. We deﬁne (a)b to be the least n such that either a = 0 or pbn+2 does not divide a. (There always is such an n.) Observe that

0, (a)b is one less than the exponent of (0)b = 0, and for a = pb in the prime factorization of a (but not less than 0). Hence for b ≤ m, ( a0 , . . . , am )b = ab . To prove representability we use the least-zero operator. Let R = { a, b, n | a = 0 or pbn+2 does not divide a}. Then (a)b = μn[K R (a, b, n) = 0], where R is the complement of R. Since the method used in the above proof will be useful elsewhere as well, we here state it separately:

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THEOREM 33N Assume that R is a representable relation such that for every a there is some n such that a , n ∈ R. Then the function f deﬁned by f ( a ) = the least n such that a , n ∈ R is representable. PROOF.

f ( a ) = μn[K R ( a , n) = 0].

We will later use the notation f ( a ) = μn[ a , n ∈ R]. 10. Say that b is a sequence number iff for some m ≥ −1 and some a0 , . . . , a m , b = a0 , . . . , am . (When m = −1 we get = 1.) Then the set of sequence numbers is representable. PROOF.

Exercise 5.

11. There is a representable function lh such that lh a0 , . . . , am = m + 1. (Here “lh” stands for “length.” For example, lh 72 = 2.) PROOF. We deﬁne lh a to be the least n such that either a = 0 or pn does not divide a. This works. 12. There is a representable function (whose value at a, b is called the restriction of a to b, written a b) such that for any b ≤ m + 1,

a0 , . . . , am b = a0 , . . . , ab−1 . PROOF. Let a b be the least n such that either a = 0 or both n =

0 and for any j < b, any k < a p kj divides a ⇒ p kj divides n. This works.

13. (Primitive recursion) With a (k + 1)-place function f we associate another function f such that f (a, b1 , . . . , bk ) encodes the values of f ( j, b1 , . . . , bk ) for all j < a. Speciﬁcally, let = f (0, b), . . . , f (a − 1, b). f (a, b) = = 1, encoding the ﬁrst zero values of For example, f (0, b) In any case, f (a, b) is a sequence number of f . f (1, b) = f (0, b). length a, encoding the ﬁrst a values of f .

222

A Mathematical Introduction to Logic Now suppose we are given a (k + 2)-place function g. There exists a unique function f satisfying = g( f (a, b), a, b). f (a, b) For example, = g( , 0, b), f (0, b) = g( f (0, b), 1, b). f (1, b) (The existence and uniqueness of this f should be intuitively clear. For a proof, we can apply the recursion theorem of Section 1.4, obtaining ﬁrst f and then extracting f .) THEOREM 33P Let g be a (k + 2)-place function and let f be the unique (k + 1)-place function such that for all a and (k-tuples) b, = g( f (a, b), a, b). f (a, b) If g is representable, then so is f . PROOF. First we claim that f is representable. This follows from the fact that = the least s such that s is a sequence number of f (a, b) length a and for i less than a, (s)i = g(s i, i, b). It then follows that f is representable, since a, b) = g( f (a, b), f (a, b) and the functions on the right are representable.

Actually the phrase “primitive recursion” is more commonly applied to a simpler version of this, given in Exercise 8. 14. For a representable function F, the function whose value at a, b is & F(i, b) i

is also representable. Similarly with in place of . (For a = 0, we use the standard conventions: The empty product — the product of no numbers — is 1, and the empty sum is 0.) PROOF.

Call this function G; then = 1, G(0, b) · G(a, b). G(a + 1, b) = F(a, b)

Apply Exercise 8. 15. Deﬁne the concatenation of a and b, a ∗ b, by & (b) +1 a∗b =a· pi+lhi a . i

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This is a representable function of a and b, and

a1 , . . . , am ∗ b1 , . . . , bn = a1 , . . . , am , b1 , . . . , bn . The concatenation operation has the further property of being associative on sequence numbers. 16. We will also want a “capital asterisk” operation. Let

∗i

f (0) ∗ f (1) ∗ · · · ∗ f (a − 1).

For a representable function F, the function whose value at a, b is is representable. ∗i

= = 1 and ∗i<0 F(i, b) = ∗i

So this is just like catalog item 14.

Exercises 1. Show that in the structure (N; ·, E) we can deﬁne the addition relation { m, n, m + n | m, n in N}. Conclude that in this structure {0}, the ordering relation <, and the successor relation { n, S(n) | n ∈ N} are deﬁnable. (Remark: This result can be strengthened by replacing the structure (N; ·, E) by simply (N; E). The multiplication relation is deﬁnable here, by exploiting one of the laws of exponents: (d a )b = d ab .) 2. Prove Theorem 33C, stating that true (in N) quantiﬁer-free sentences are theorems of A E . (See the outline given there.) 3. A theory T (in a language with 0 and S) is called ω-complete iff for any formula ϕ and variable x, if ϕSxn 0 belongs to T for every natural number n, then ∀ xϕ belongs to T . Show that if T is a consistent ω-complete theory in the language of N and if A E ⊆ T , then T = Th N. 4. Show that in the proof preceding Theorem 33L, formula (4) is logically implied by the set consisting of formulas (1), (2), and (3). 5. Show that the set of sequence numbers is representable (catalog item 10). 6. Is 3 a sequence number? What is lh 3? Find (1∗3)∗6 and 1∗(3∗6). 7. Establish the following facts: (a) a + 1 < pa . (b) (b)k ≤ b; equality holds iff b = 0. (c) lh a ≤ a; equality holds iff a = 0. (d) a i ≤ a. (e) lh(a i) is the smaller of i and lh a.

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A Mathematical Introduction to Logic 8. Let g and h be representable functions, and assume that f (0, b) = g(b), f (a + 1, b) = h( f (a, b), a, b). Show that f is representable. 9. Show that there is a representable function f such that for every n, a0 , . . . , an , f ( a0 , . . . , an ) = an . (For example, f (72) = 1 and f (750) = 2.) 10. Assume that R is a representable representable functions. Show that g( a) f ( a) = h( a)

relation and that g and h are f is representable, where if a ∈ R, if a ∈ / R.

11. (Monotone recursion) Assume that R is a representable binary relation on N. Let C be the smallest subset of N (i.e., the intersection of all subsets) such that for all n, a0 , . . . , an−1 , b, a0 , . . . , an−1 , b ∈ R & ai ∈ C (for all i < n) ⇒ b ∈ C. Further assume that (1) for all n, a0 , . . . , an−1 , b, a0 , . . . , an−1 , b ∈ R ⇒ ai < b (for all i < n), and (2) there is a representable function f such that for all n, a0 , . . . , an−1 , b, a0 , . . . , an−1 , b ∈ R ⇒ n < f (b) Show that C is representable. (C is, in a sense, generated by R. C=

∅ in general because if , b ∈ R, then b ∈ C.)

SECTION 3.4 Arithmetization of Syntax In this section we intend to develop two themes: 1. Certain assertions about wffs can be converted into assertions about natural numbers (by assigning numbers to expressions). 2. These (English) assertions about natural numbers can in many cases be translated into the formal language. And the theory Cn A E is strong enough to prove many of the translations so obtained. This will give us the ability to construct formulas that, by expressing facts about numbers, indirectly express facts about formulas (even about themselves!). Such an ability will be exploited in Section 3.5 to obtain results of undeﬁnability and undecidability.

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Godel ¨ Numbers We ﬁrst want to assign numbers to expressions of the formal language. Recall that the symbols of our language are those listed in Table IX. TABLE IX Parameters 0. 2. 4. 6. 8. 10. 12.

∀ 0 S

< +

· E

Logical symbols 1. 3. 5. 7. 9. 11. 13.

( ) ¬ →

= v1 v2 , etc.

There is a function h assigning to each symbol the integer listed to its left. Thus h(∀) = 0, h(0) = 2, and h(vi ) = 9 + 2i. In order to make our subsequent work more widely applicable, we will assume only that we have some language with 0 and S which is recursively numbered. By this we mean that we have a one-to-one function h from the parameters of that language into the even numbers such that the two relations { k, m | k is the value of h at some m-place predicate parameter} and { k, m | k is the value of h at some m-place function symbol} are both representable in Cn A E . Of course in the case of the language of N these sets are even ﬁnite. The ﬁrst set is { 6, 2 } and the second is { 2, 0 , 4, 1 , 8, 2 , 10, 2 , 12, 2 }. We deﬁne h on the logical symbols as before; thus h(s) is an odd number for each logical symbol s. For an expression ε = s0 · · · sn of the language we deﬁne its G¨odel number, (ε), by (s0 · · · sn ) = h(s0 ), . . . , h(sn ). For example, using our original function h for the language of N, we obtain (∃ v3 v3 = 0) = (( ¬ ∀ v3 ( ¬ =v3 0))) = 1, 5, 0, 15, 1, 5, 9, 15, 2, 3, 3 = 22 · 36 · 51 · 716 · 112 · 136 · 1710 · 1916 · 233 · 294 · 314 .

226

A Mathematical Introduction to Logic This is a large number, being of the order of 1.3 × 1075 . To a set of expressions we assign the set = {(ε) | ε ∈ } of G¨odel numbers. To a sequence α0 , . . . , αn of expressions (such as a deduction), we assign the number G( α0 , . . . , αn ) = α0 , . . . , αn . We now proceed to show that various relations and functions having to do with G¨odel numbers are representable in Cn A E (and hence are recursive). As in the preceding section, whenever we say that a relation or function is representable (without specifying a theory) we mean that it is representable in the theory Cn A E . We will make use of certain abbreviations in the language we use (i.e., English, although it is coming to differ more and more from what one ordinarily thinks of as English). For “there is a number a such that” we write “∃a.” In the same spirit, “∃a, b < c” means “there are numbers a and b both of which are less than c such that.” Similarly, we may employ “∀.” We would not have dared to employ such abbreviations in Chapter 2, for fear of creating confusion between the formal language and the meta-language (English). But by now we trust the reader to avoid such erroneous ways. 1. The set of G¨odel numbers of variables is representable. PROOF. It is {a | (∃b < a)a = 11 + 2b}. It follows from results of the preceding section that this is a representable set. 2. The set of G¨odel numbers of terms is representable. PROOF. The set of terms was deﬁned inductively. And terms were built up from constituents with smaller G¨odel numbers. We will treat this case in some detail, since it is typical of the argument used for inductively deﬁned relations. Let f be the characteristic function of the set of G¨odel numbers of terms. From the deﬁnition of “term” we obtain ⎧ 1 if a is the G¨odel number of a variable, ⎪ ⎪ ⎪ ⎪ 1 if (∃i < , ∃k < a) [i is a sequence number ⎪ ⎪ ⎨ & (∀ j < lh i) f ((i) j ) = 1 & k is the value of f (a) = h at some (lh i)-place function symbol & ⎪ ⎪ ⎪ ⎪ a = k ∗ ∗ j

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need an upper bound on i that depends in some representable way on a. The claim is that we can take i < a a lh a . To see this, suppose that a = st1 · · · tn (where s is an n-place function symbol and t1 , . . . , tn are terms). Then we want to take i = t1 , . . . , tn . How big could this be, in terms of a? We have the bounds: t +1

n i = 2t1 +1 · · · pn−1 a a ≤ 2 · · · pn−1 a < 2a · · · plh a−1 because n = lh i < lh a (a)lh a−1 +1 a a ≤ a · · · a (lh a times) because a = 2(a)0 +1 · · · plh a−1 ≥ plh a−1 = (a a )lh a = a a lh a

So in the above equation for f , we replace by a a lh a . Although the right side of this equation refers to f , it refers only to f ((i) j ), where (i) j < a. This feature permits us to apply primitive recursion. f (a) = g( f (a), a), where

g(s, a) =

⎧ 1 ⎪ ⎪ ⎪ ⎪ 1 ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

0

if a is the G¨odel number of a variable, if (∃i < a a lh a , ∃k < a) [i is a sequence number & (∀ j < lh i)(s)(i) j = 1 & k is the value of h at some (lh i)-place function symbol & a = k ∗ ∗ j

For if in this equation we set s equal to f (a), then (s)(i) j = f ((i) j ) for (i) j < a. Hence by Theorem 33P, f is representable provided that g is. It remains to show that g is representable. But this is straightforward, by using results of the preceding section. Brieﬂy, the graph of g is the union of three relations, corresponding to the three clauses in the above equation. Each of the three is obtained from equality and other representable relations by bounded quantiﬁcation and the substitution of representable functions. 3. The set of G¨odel numbers of atomic formulas is representable. PROOF. a is the G¨odel number of an atomic formula iff (∃i < a a lh a , ∃k < a) [i is a sequence number & (∀ j < lh i)(i) j is the G¨odel number of a term & k is the value of h at some (lh i)-place predicate symbol & a = k ∗ ∗ j

228

A Mathematical Introduction to Logic PROOF. The wffs were inductively deﬁned. Let f be the characteristic function of the set, then ⎧ 1 if a is the G¨odel number of an atomic formula, ⎪ ⎪ ⎪ ⎪ 1 if (∃i < a)[a = h((), h(¬) ∗ i ∗ h()) ⎪ ⎪ ⎪ ⎪ & f (i) = 1], ⎪ ⎪ ⎨ 1 if (∃i, j < a)[a = h(() ∗ i ∗ h(→) ∗ j ∗ h()) f (a) = & f (i) = f ( j) = 1], ⎪ ⎪ ⎪ ⎪ (∃i, j < a)[a = h(∀) ∗ i ∗ j & i is the G¨odel 1 if ⎪ ⎪ ⎪ ⎪ number of a variable and f ( j) = 1], ⎪ ⎪ ⎩ 0 otherwise. By the same argument used for the set of G¨odel numbers of terms, we have the representability of f . 5. There is a representable function Sb such that for a term or formula α, variable x, and term t, Sb(α, x, t) = αtx . PROOF. We will need to deﬁne Sb(a, b, c) making use of values Sb(i, b, c) where i < a. As in the case of catalog item 2 (the characteristic function of the set of terms), it will then be possible to show that both Sb and Sb are representable. The function Sb is described by the following six clauses (i)–(vi): (i) If a is the G¨odel number of a variable and a = b then Sb(a, b, c) = c. (ii) If (∃i < a a lh a , ∃k < a)[i is a sequence number & (∀ j < lh i)(i) j is the G¨odel number of a term & k is the value of h at some (lh i)-place function or predicate symbol & a = k ∗ ∗ j

h((), h(¬) ∗ i ∗ h())] then Sb(a, b, c) = h((), h(¬) ∗ Sb(i, b, c) ∗ h()) for that i. (iv) If (∃i, j < a)[i and j are G¨odel numbers of wffs & a =

h(() ∗ i ∗ h(→) ∗ j ∗ h())] then Sb(a, b, c) = h(() ∗ Sb(i, b, c) ∗ h(→) ∗ Sb( j, b, c) ∗ h()) for that i and j.

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(v) If (∃i, j < a)[i is the G¨odel number of a variable & i =

b & j is the G¨odel number of a wff & a = h(∀) ∗ i ∗ j] then Sb(a, b, c) = h(∀) ∗ i ∗ Sb( j, b, c) for that i and j. (vi) If none of the above conditions on a and b are met (where we ignore the displayed equation for Sb(a, b, c)) then Sb(a, b, c) = a. Then the function Sb is obtained by primitive recursion Sb(a, b, c) = G(Sb(a, b, c), a, b, c) where G is a 4-place function. The graph of G is the union of six 5-ary relations G = R1 ∪ R 2 ∪ R 3 ∪ R 4 ∪ R 5 ∪ R 6 corresponding to the six clauses above. The ﬁrst of the six is R1 = { s, a, b, c, d | a is the G¨odel number of a variable & a = b & d = c}. The second one is R2 = { s, a, b, c, d | (∃i < a a lh a , ∃k < a)[i is a sequence number & (∀ j < lh i)(i) j is the G¨odel number of a term & k is the value of h at some (lh i)-place function or predicate symbol & a = k ∗ ∗ j

Call this function f ; then f (0) = h(0), f (n + 1) = h(S) ∗ f (n). Apply Exercise 8 of the preceding section.

7. There is a representable relation Fr such that for a term or formula α and a variable x, α, x ∈ Fr ⇔ x occurs free in α.

230

A Mathematical Introduction to Logic PROOF.

a, b ∈ Fr ⇔ Sb(a, b, 0) =

a.

8. The set of G¨odel numbers of sentences is representable. PROOF. a is the G¨odel number of a sentence iff a is the G¨odel number of a formula and for any b < a, if b is the G¨odel number of a variable then a, b ∈ / Fr. 9. There is a representable relation Sbl such that for a formula α, variable x, and term t, a, x, t ∈ Sbl iff t is substitutable for x in α. PROOF.

Exercise 1.

10. The relation Gen, where a, b ∈ Gen iff a is the G¨odel number of a formula and b is the G¨odel number of a generalization of that formula, is representable. PROOF. a, b ∈ Gen iff a = b or (∃i, j < b)[i is the G¨odel number of a variable and a, j ∈ Gen and b = (h(∀) ∗ i ∗ j]. Apply the usual argument to the characteristic function of Gen. 11. The set of G¨odel numbers of tautologies is representable. The set of tautologies is informally decidable since we can use the method of truth tables. To obtain representability, we recast truth tables in terms of G¨odel numbers. There are several preliminary steps: 11.1 The relation R, such that a, b ∈ R iff a is the G¨odel number of a formula α and b is the G¨odel number of a prime constituent of α, is representable. PROOF. a, b ∈ R ⇔ a is the G¨odel number of a formula and one of the following: (i) a = b & (a)0 =

h((). (ii) (∃i < a)[a = h((), h(¬) ∗ i ∗ (h()) and i, b ∈ R]. (iii) The analogue to (ii) but with →. Apply the usual argument to the characteristic function of R. 11.2 There is a representable function P such that for a formula α, P(α) = β1 , . . . , βn , the list of G¨odel numbers of prime constituents of α, in numerical order. PROOF. First deﬁne a function g for locating the next prime constituent in a after y (where a is the formula α for which a = α). g(a, y) = the least n such that either n = a + 1 or both y < n and a, n ∈ R. Next deﬁne a function h such that h(a, n) gives the (n +1)st prime constituent of a (if there are that many): h(a, 0) = g(a, 0)

h(a, n + 1) = g(a, h(a, n)).

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231

Finally, let P(a) = ∗i

(P(α))i , e. This is a representable condition on v and α. For example, if P(α) = β0 , . . . , βn , then v =

β0 , e0 , . . . , βn , en , where each ei is 0 or 1. We will later need an upper bound for v in terms of α. The largest v is obtained when each ei is 1. Also βi ≤ α, so that v ≤

α, 1, . . . , α, 1 = ∗i

α, 1. 11.4 There is a representable relation Tr such that for a formula α and a v which encodes a truth assignment for α (or more), α, v ∈ Tr iff that truth assignment satisﬁes α. PROOF.

Exercise 2.

Finally, α is a tautology iff α is a formula and for every v encoding a truth assignment for α, α, v ∈ Tr. The (English) quantiﬁer on v can be bounded by a representable function of α, as explained in 11.3. 12. The set of G¨odel numbers of formulas of the form ∀ x ϕ → ϕtx , where t is a term substitutable for the variable x in ϕ, is representable. PROOF. α is of this form iff (∃ wff ϕ < α)(∃ variable x < α)(∃ term t < α)[t is substitutable for x in ϕ and α = ∀ x ϕ → ϕtx ]. Here “ϕ < α” means that ϕ < α. This is easily rewritten in terms of G¨odel numbers: a belongs to the set iff (∃ f < a)(∃x < a)(∃t < a) [ f is the G¨odel number of a formula & x is the G¨odel number of a variable & t is the G¨odel number of term and f, x, t ∈ Sbl & a = h( ( ), h(∀) ∗ x ∗ f ∗ h(→) ∗ Sb( f, x, t) ∗ h())]. 13. The set of G¨odel numbers of formulas of the form ∀ x(α → β) → ∀ x α → ∀ x β is representable. PROOF. γ is of this form iff (∃ variable x < γ )(∃ formulas α, β < γ ) [γ = ∀ x(α → β) → ∀ x α → ∀ x β]. This is easily rewritten in terms of G¨odel numbers, as in 12. 14. The set of G¨odel numbers of formulas of the form α → ∀ x α, where x does not occur free in α, is representable. PROOF.

Similar to 13.

15. The set of G¨odel numbers of formulas of the form x = x is representable.

232

A Mathematical Introduction to Logic PROOF.

Similar to 13.

16. The set of G¨odel numbers of formulas of the form x = y→α→α , where α is atomic and α is obtained from α by replacing x at zero or more places by y, is representable. PROOF. This is similar to 13, except for the relation of “partial substitution.” Let a, b, x, y ∈ Psb iff x and y are G¨odel numbers of variables, a is the G¨odel number of an atomic formula, b is a sequence number of length lh a, and for all j < lh a, either (a) j = (b) j or (a) j = x and (b) j = y. This relation is representable. 17. The set of G¨odel numbers of logical axioms is representable. PROOF. α is a logical axiom iff ∃β ≤ α such that α is a generalization of β and β is in one of the sets in items 11–16. 18. For a ﬁnite set A of formulas, {G(D) | D is a deduction from A} is representable. In fact it is enough here for A to be representable. PROOF. A number d belongs to this set iff d is a sequence number of positive length and for every i less than lh d, either 1. (d)i ∈ A, 2. (d)i is the G¨odel number of a logical axiom, or 3. (∃ j, k < i)[(d) j = h(() ∗ (d)k ∗ h(→)∗(d)i ∗ h())]. This is representable whenever A is, as is certainly the case for ﬁnite A. 19. Any recursive relation is representable in Cn A E . PROOF. Recall that the relation R is recursive iff there is some ﬁnite consistent set A of sentences such that some formula ρ represents R in Cn A. (There is no loss of generality in assuming that the language has only ﬁnitely many parameters: those in the ﬁnite set A, those in ρ, and 0, S, and ∀.) In the case of a unary relation R, we have that a ∈ R iff the least D which is a deduction from A of either ρ(Sa 0) or ¬ ρ(Sa 0) is, in fact, a deduction of the former. More formally, a ∈ R iff the last component of f (a) is ρ(Sa 0), where f (a) = the least d such that d is in the set of item 18 and the last component of d is either ρ(Sa 0) or ¬ ρ(Sa 0). For this (ﬁxed) ρ, there always is such a d. Since the converse to item 19 is immediate, we have THEOREM 34A A relation is recursive iff it is representable in the theory Cn A E .

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Henceforth we will usually use the word “recursive” in preference to “representable.” COROLLARY 34B

Any recursive relation is deﬁnable in N.

20. Now suppose we have a set A of sentences such that A is recursive. Then Cn A need not be recursive (as we will show in the next section). But we do have a way of deﬁning Cn A from A: a ∈ Cn A

iff

∃d[d is the number of a deduction from A and the last component of d is a and a is the G¨odel number of a sentence]

The part in square brackets is recursive, by the proof to item 18. But we cannot in general put any bound on the number d. The best we can say is that Cn A is the domain of a recursive relation (or, as we will say later, is recursively enumerable). Item 20 will play a key role our subsequent work. In particular, it will later be restated as Theorem 35I. 21. If A is recursive and Cn A is a complete theory, then Cn A is recursive. In other words, a complete recursively axiomatizable theory is recursive. This is the analogue to Corollary 25G, which asserts that a complete axiomatizable theory is decidable. The proof is essentially unchanged. Let (in the consistent case) g(s) = the least d such that s is not the G¨odel number of a sentence, or d is in the set of item 18 and the last component of d is either s or is h((), h(¬) ∗ s ∗ h()). Thus g(σ ) is G of the least deduction of σ or ( ¬ σ ) from A. And s ∈ Cn A iff s > 0 and the last component of g(s) is s. At this point we might reconsider the plausibility of Church’s thesis. Suppose that the relation R is decidable. Then there is a ﬁnite list of explicit instructions (a program) for the decision procedure. The procedure itself will presumably consist of certain atomic steps, which are then performed repeatedly. (The reader familiar with computer programming will know that a short program can still require much time for its execution, but some commands will be utilized over and over.) Any one atomic step is presumably very simple. By devices akin to G¨odel numbering, we can mirror the decision procedure in the integers. The characteristic function of R can then be put in the form

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A Mathematical Introduction to Logic K R ( a ) = U [the least s such that (i) (s)0 encodes the input a ; (ii) for all positive i < lh s, (s)i is obtained from (s)i−1 by performance of the applicable atomic step; (iii) the last component of s describes a terminal situation, at which the computation is completed], where U (the upshot function) is some simple function that extracts from the last component of s the answer (afﬁrmative or negative). The recursiveness of R is now reduced to the recursiveness of U and of the relations indicated in (i), (ii), and (iii). In special cases, such as decision procedures provided by the register machines of Section 3.6, the recursiveness of these components is easily veriﬁed. It seems most improbable that any decision procedure will ever be regarded as effective and yet will have components that are nonrecursive. For example, in (ii), it seems that it ought to be possible to make each atomic step extremely simple, and in particular to make each one recursive.

Exercises 1. Supply a proof for item 9 of this section. 2. Supply a proof for item 11.4 of this section. 3. Use Exercise 11 of Section 3.3 to give a new proof that the set of G¨odel numbers of terms is representable (item 2). 4. Let T be a consistent recursively axiomatizable theory (in a recursively numbered language with 0 and S). Show that any relation representable in T must be recursive.

SECTION 3.5 Incompleteness and Undecidability In this section we reap the rewards of our work in Sections 3.3 and 3.4. We have assigned G¨odel numbers to expressions, and we have shown that certain intuitively decidable relations on N (related to syntactical notions about expressions) are representable in Cn A E . Throughout this section we assume that the language in question is the language of N. (This affects the meaning of “Cn” and “theory.”) FIXED-POINT LEMMA Given any formula β in which only v1 occurs free, we can ﬁnd a sentence σ such that A E ! [σ ↔ β(Sσ 0)]. We can think of σ as indirectly saying, “β is true of me.” Actually, of course, σ doesn’t say anything; it’s just a string of symbols. And even when translated into English according to the intended

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structure N, it then talks of numbers and their successors and products and so forth. It is only by virtue of our having associated numbers with expressions that we can think of σ as referring to a formula, in this case to σ itself. PROOF. Let θ (v1 , v2 , v3 ) functionally represent in Cn A E a function whose value at α, n is (α(Sn 0)). (See items 5 and 6 in Section 3.4.) First consider the formula ∀ v3 [θ(v1 , v1 , v3 ) → β(v3 )].

(1)

(We may suppose v3 is substitutable for v1 in β. The above formula has only v1 free. It deﬁnes in N a set to which α belongs iff (α(Sα 0)) is in the set deﬁned by β.) Let q be the G¨odel number of (1). Let σ be ∀ v3 [θ(Sq 0, Sq 0, v3 ) → β(v3 )]. Thus σ is obtained from (1) by replacing v1 , by Sq 0. Notice that σ does assert (under N) that σ is in the set deﬁned by β. But we must check that σ ↔ β(Sσ 0)

(2)

is a consequence of A E . Because θ functionally represents a function whose value at q, q is σ , we have A E ! ∀ v3 [θ(Sq 0, Sq 0, v3 ) ↔ v3 = Sσ 0].

(3)

We can obtain (2) as follows: (→) It is clear (by looking at σ ) that σ ! θ(Sq 0, Sq 0, Sσ ) → β(Sσ 0). And, by (3), A E ! θ(Sq 0, Sq 0, Sσ 0). Hence A E ; σ ! β(Sσ 0), which gives half of (2). (←) The sentence in (3) implies β(Sσ 0) → [∀ v3 (θ (Sq 0, Sq 0, v3 ) → β(v3 ))]. But the part in square brackets is just σ .

(Sometimes the notation % σ & is used for Sσ 0. In this notation, the ﬁxed-point lemma states that A E ! (σ ↔ β(% σ & )).) Our ﬁrst application of this lemma does not concern the subtheory Cn A E , and requires only the weaker fact that |=N [σ ↔ β(Sσ 0)].

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A Mathematical Introduction to Logic TARSKI UNDEFINABILITY THEOREM (1933) able in N.

The set Th N is not deﬁn-

PROOF. Consider any formula β (which you suspect might deﬁne Th N). By the ﬁxed-point lemma (applied to ¬ β) we have a sentence σ such that |=N [σ ↔ ¬ β(Sσ 0)]. (If β did deﬁne Th N, then σ would indirectly say “I am false.”) Then |=N σ ⇐⇒ |=N β(Sσ 0), so either σ is true but (its G¨odel number is) not in the set β deﬁnes, or it is false and in that set. Either way σ shows that β cannot deﬁne Th N. The above theorem gives at once the undecidability of the theory of N: COROLLARY 35A PROOF.

Th N is not recursive.

Any recursive set is (by Corollary 34B) deﬁnable in N.

GODEL ¨ INCOMPLETENESS THEOREM (1931) If A ⊆ Th N and A is recursive, then Cn A is not a complete theory. Thus there is no complete recursive axiomatization of Th N. PROOF. Since A ⊆ Th N, we have Cn A ⊆ Th N. If Cn A is a complete theory, then equality holds. But if Cn A is a complete theory, then Cn A is recursive (item 21 of the preceding section). And by the above corollary, Th N is not recursive. In particular, Cn A E is not a complete theory and so is not equal to Th N. And the incompleteness would not be eliminated by the addition of any recursive set of true axioms. (By a recursive set of sentences we mean of course a set for which is recursive.) We can extract more information from the proof of G¨odel’s theorem. Suppose we have a particular recursive A ⊆ Th N in mind. Then by item 20 in Section 3.4 we can ﬁnd a formula β that deﬁnes Cn A in N. The sentence σ produced by the proof of Tarski’s theorem is (as we noted there) a true sentence not in Cn A. This sentence asserts that σ does not belong to the set deﬁned by β, i.e., it indirectly says, “I am not a theorem of A.” Thus A σ , and of course A ¬ σ as well. This way of viewing the proof is closer to G¨odel’s original proof, which did not involve a detour through Tarski’s theorem. For that matter, G¨odel’s statement of the theorem did not involve Th N; we have taken some liberties in the labeling of theorems.

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Next we need a lemma which says (roughly) that one can add one new axiom (and hence ﬁnitely many new axioms) to a recursive theory without losing the property of recursiveness. LEMMA 35B PROOF.

If Cn is recursive, then Cn(; τ ) is recursive.

α ∈ Cn(; τ ) ⇔ (τ → α) ∈ Cn . Thus

a ∈ Cn(; τ ) ⇐⇒ a is the G¨odel number of a sentence and h(() ∗ τ ∗ h(→) ∗ a ∗ h()) is in Cn . This is recursive by the results of the preceding sections.

THEOREM 35C (STRONG UNDECIDABILITY OF CN AE ) Let T be a theory such that T ∪ A E is consistent. Then T is not recursive. (Notice that because throughout this section the language in question is the language of N, the word “theory” here means “theory in the language of N.”) PROOF. Let T be the theory Cn(T ∪ A E ). If T is recursive, then since A E is ﬁnite we can conclude by the above lemma that T is also recursive. Suppose, then, that T is recursive and so is represented in Cn A E by some formula β. From the ﬁxed-point lemma we get a sentence σ such that A E ! [σ ↔ ¬ β(Sσ 0)].

(∗)

(Indirectly σ asserts, “I am not in T .”) σ ∈ / T ⇒ ⇒ ⇒ ⇒

σ ∈ / T A E ! ¬ β(Sσ 0) AE ! σ σ ∈ T .

by (∗)

So we get σ ∈ T . But this, too, is untenable: σ ∈ T ⇒ ⇒ ⇒ ⇒

σ ∈ T A E ! β(Sσ 0) by (∗) AE ! ¬ σ ( ¬ σ ) ∈ T ,

which contradicts the consistency of T .

COROLLARY 35D Assume that is recursive and ∪ A E is consistent. Then Cn is not a complete theory. PROOF. A complete recursively axiomatizable theory is recursive (item 21 of Section 3.4). But Cn is not recursive, by the above theorem. This corollary is G¨odel’s incompleteness theorem again, but with truth in N replaced by consistency with A E .

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A Mathematical Introduction to Logic CHURCH’S THEOREM (1936) The set of G¨odel numbers of valid sentences (in the language of N) is not recursive. PROOF. In the strong undecidability of Cn A E , take T to be the smallest theory in the language, the set of valid sentences. The set of G¨odel numbers of valid wffs is not recursive either, lest the set of valid sentences be recursive. This proof applies to the language of N. For a language with more parameters, the set of valid sentences is still nonrecursive (lest its intersection with the language of N be recursive). Actually it is enough for the language to contain at least one two-place predicate symbol. (See Corollary 37G.) On the other hand, some lower bound on the language is needed. If the language has ∀ as its only parameter (the language of equality), then the set of valid formulas is decidable. (See Exercise 6.) More generally, it is known that if the only parameters are ∀ and oneplace predicate symbols, then the set of valid formulas is decidable.

Recursive Enumerability A relation on the natural numbers is said to be recursively enumerable iff it is of the form { a | ∃b a , b ∈ Q} with Q recursive. Recursively enumerable relations play an important role in logic. They constitute the formal counterpart to the effectively enumerable relations (as will be explained presently). (The standard abbreviation for “recursively enumerable” is “r.e.” When the term “computable” is used instead of “recursive,” then one speaks of computably enumerable — abbreviated c.e. — relations.) Recursively enumerable relations are — like the recursive relations — deﬁnable in N. If ϕ(v1 , v2 ) deﬁnes in N a binary relation Q, then ∃ v2 ϕ(v1 , v2 ) deﬁnes {a | ∃b a, b ∈ Q}. THEOREM 35E The following conditions on an m-ary relation R are equivalent: 1. R is recursively enumerable. 2. R is the domain of some recursive relation Q. 3. For some recursive (m + 1)-ary relation Q, R = { a1 , . . . , am | ∃b a1 , . . . , am , b ∈ Q}. 4. For some recursive (m + n)-ary relation Q, R = { a1 , . . . , am | ∃b1 , . . . , bn a1 , . . . , am , b1 , . . . , bn ∈ Q}. PROOF. By deﬁnition 1 and 3 are equivalent. Also 2 and 3 are equivalent by our deﬁnition (in Chapter 0) of domain and (m +1)-tuple.

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Clearly 3 implies 4. So we have only to show that 4 implies 3. This is because ∃b1 , . . . , bn a1 , . . . , am , b1 , . . . , bn ∈ Q iff ∃c a1 , . . . , am , (c)0 , . . . , (c)n−1 ∈ Q and { a1 . . . , am , c | a1 , . . . , am , (c)0 , . . . , (c)n−1 ∈ Q} is recursive whenever Q is recursive. (Here we have used our sequence decoding function to collapse a string of quantiﬁers into a single one.) By part 4 of this theorem, R is recursively enumerable iff it is deﬁnable in N by a formula ∃ x1 · · · ∃ xn ϕ, where ϕ is numeralwise determined by A E . In fact, we can require here that ϕ be quantiﬁer-free; this result was proved in 1961 (with exponentiation) and in 1970 (without exponentiation). The proofs involve some number theory; we will omit them here. Notice that any recursive relation is also recursively enumerable. For if R is recursive, then it is deﬁned in N by a formula ∃ x1 · · · ∃ xn ϕ, where ϕ is numeralwise determined by A E and x1 , . . . , xn do not occur in ϕ. THEOREM 35F A relation is recursive iff both it and its complement are recursively enumerable. This is the formal counterpart to the fact (cf. Theorem 17F) that a relation is decidable iff both it and its complement can be effectively enumerated. PROOF. If a relation is recursive, then so is its complement, whence both are recursively enumerable. Conversely, suppose that both P and its complement are recursively enumerable; thus for any a , a ∈ P ⇔ ∃b a , b ∈ Q a ∈ / P ⇔ ∃b a , b ∈ R for some recursive Q and R. Let f ( a ) = the least b such that either a , b ∈ Q or a , b ∈ R. Such a number b always exists, and f is recursive. Finally, a ∈ P ⇔ a , f ( a ) ∈ Q, so P is recursive.

The recursively enumerable relations constitute the formal counterpart of the effectively enumerable relations. For we have the following

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A Mathematical Introduction to Logic informal result, paralleling a characterization of recursive enumerability given by Theorem 35E.

LEMMA 35G A relation is effectively enumerable iff it is the domain of a decidable relation.

PROOF. Assume that Q is effectively enumerated by some procedure. Then a ∈ Q iff ∃n[ a appears in the enumeration in n steps]. The relation deﬁned in square brackets is decidable and has domain Q. Conversely, to enumerate { a, b | ∃n a, b, n ∈ R} for decidable R, we check to see if (m)0 , (m)1 , (m)2 ∈ R for m = 0, 1, 2, . . . . Whenever the answer is afﬁrmative, we place (m)0 , (m)1 on the output list.

COROLLARY 35H (CHURCH’S THESIS, SECOND FORM) A relation is effectively enumerable iff it is recursively enumerable.

PROOF. By identifying the class of decidable relations with the class of recursive relations, we automatically identify the domains of decidable relations with domains of recursive relations. The second form of Church’s thesis is, in fact, equivalent to the ﬁrst form. To prove the ﬁrst form from the second, we use Theorems 35F and 17F. We have already shown that a recursively axiomatizable theory is recursively enumerable, but using different words. We restate the result here, as it indicates the role recursive enumerability plays in logic. THEOREM 35I If A is a set of sentences such that A is recursive, then Cn A is recursively enumerable. PROOF.

Item 20 of Section 3.4.

In particular, Cn A E is recursively enumerable, but (by Theorem 35C) it is not recursive. In the next section, we will see other examples of recursively enumerable sets that are not recursive. This theorem is the precise counterpart of the informal fact that a theory with a decidable set of axioms is effectively enumerable (Corollaries 25F and 26I). It indicates the gap between what is provable in an axiomatic theory and what is true in the intended structure. With a recursive set of axioms, all we can possibly obtain is a recursively enumerable set of consequences. But by Tarski’s theorem, Th N is not even deﬁnable in N, much less recursively enumerable. Even if we expand the language or add new axioms, the same phenomenon is present. As long as we can recursively distinguish deductions from nondeductions, the set of theorems can be only recursively enumerable. For example, the set of sentences of number theory provable in your favorite system of axiomatic set theory is recursively

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enumerable. Furthermore, this set includes A E and is consistent (unless you have very strange favorites). It follows that this set theory is nonrecursive and incomplete. (This topic is discussed more carefully in Section 3.7.)

Weak Representability Consider a recursively enumerable set Q, where a ∈ Q ⇔ ∃b a, b ∈ R for recursive R. We know there is a formula ρ that represents R in Cn A E . Consequently, the formula ∃v2 ρ deﬁnes Q in N. This formula cannot represent Q in Cn A E unless Q is recursive. But it can come halfway. a∈Q ⇒ ⇒ ⇒ a∈ / Q ⇒ ⇒ ⇒

a, b ∈ R for some b A E ! ρ(Sa 0, Sb 0) for some b A E ! ∃ v2 ρ(Sa 0, v2 ) a, b ∈ / R for all b A E ! ¬ ρ(Sa 0, Sb 0) for all b A E ∃ v2 ρ(Sa 0, v2 )

The last step is justiﬁed by the fact that if A E ! ¬ ϕ(Sb 0) for all b, then A E ∃ x ϕ(x). (The term ω-consistency is given to this property.) For it is impossible for ∃ x ϕ(x), ¬ ϕ(S0 0), ¬ ϕ(S1 0), . . . all to be true in N. Thus we have a ∈ Q ⇔ A E ! ∃ v2 ρ(Sa 0, v2 ). It will be convenient to formulate a deﬁnition of this half of representability. DEFINITION. Let Q be an n-ary relation on N, ψ a formula in which only v1 , . . . , vn occur free. Then ψ weakly represents Q in a theory T iff for every a1 , . . . , an in N, a1 , . . . , an ∈ Q ⇔ ψ(Sa1 0, . . . , San 0) ∈ T. Observe that if Q is representable in a consistent theory T , then Q is also weakly representable in T . THEOREM 35J A relation is weakly representable in Cn A E iff it is recursively enumerable. PROOF. We just showed that a recursively enumerable unary relation Q is weakly representable in Cn A E ; the same proof applies to n-ary Q with only notational changes. Conversely, let Q be

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A Mathematical Introduction to Logic weakly represented by ψ in Cn A E . Then a1 , . . . , an ∈ Q ⇔ ∃D [D is a deduction of ψ(Sa1 0, . . . , San 0) from the axioms A E ] ⇔ ∃d d, f (a1 , . . . , an ) ∈ P for a certain recursive function f and recursive relation P.

Arithmetical Hierarchy Deﬁne a relation on the natural numbers to be arithmetical iff it is deﬁnable in N. But some arithmetical relations are, in a sense, more deﬁnable than others. We can organize the arithmetical relations into a hierarchy according to how deﬁnable the relations are. Let 1 be the class of recursively enumerable relations; these relations are “one quantiﬁer away” from recursiveness. Extending this idea, we deﬁne the class of k relations and the class of k relations. For example, the ﬁrst few classes consist of relations of the form shown in the second column: 1 : 1 : 2 : 2 :

{ a { a { a { a

| ∃b a , b ∈ R}, | ∀b a , b ∈ R}, | ∃c∀b a , b, c ∈ R}, | ∀c∃b a , b, c ∈ R},

R recursive. R recursive. R recursive. R recursive.

In general, a relation Q is in k iff it is of the form ∈ R} a , b { a | ∀b1 ∃b2 · · · bk for a recursive relation R. Here “” is to be replaced by “∀” if k is odd and by “∃” if k is even. Similarly, Q is in k iff it has the form ∈ R} a , b { a | ∃b1 ∀b2 · · · bk for recursive R, where now “” is replaced by “∃” if k is odd and by “∀” if k is even. The classes k and k can also be deﬁned by recursion on k. 1 is the class of recursively enumerable relations. Next, a relation belongs to k iff its complement is in k . And a relation is in k+1 iff it is the domain of a relation in k . (We can even start from k = 0, by letting 0 be the class of recursive relations.) EXAMPLE. The set of G¨odel numbers of formulas numeralwise determined by A E is in 2 . PROOF. a belongs to this set iff [a is the G¨odel number of a formula α] and ∀b∃d[d is G of a deduction from A E either of α(S(b)0 0, S(b)1 0, . . .) or of the negation of this sentence]. By the technique of Section 3.4, we can show that the phrases in square brackets deﬁne recursive relations. By using the English

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counterpart to prenex form, we obtain the desired form, {a | ∀b∃d a, b, d ∈ R},

with R recursive.

One more bit of notation: Let 1 be the class of recursive relations. Then our earlier result (Theorem 35F) stating that a relation is recursive iff both it and its complement are recursively enumerable can now be stated by the equation 1 = 1 ∩ 1 . Since this equation holds, we proceed to deﬁne n for n > 1 by the analogous equation, n = n ∩ n . The following inclusions hold:

The case 1 ⊆ 1 was mentioned previously (cf. Theorem 35F); its proof hinged on the possibility of “vacuous quantiﬁcation.” The proofs of the other cases are conceptually the same. If x does not occur in ϕ, then ϕ, ∀ x ϕ, and ∃ x ϕ are all equivalent. For example, a relation in 1 is deﬁned by a formula ∃ y ϕ, where ϕ is numeralwise determined by A E . But the same relation is deﬁned by ∃ y ∀ x ϕ and ∀ x ∃ y ϕ (where x does not occur in ϕ). Hence the relation is also in 2 and 2 . It is also true that all the inclusions shown are proper inclusions, i.e., equality does not hold. But we will not prove this fact here. The inclusions are shown pictorially in Fig. 10. The class of arithmetical relations equals k k and also k k . For example, any relation in 2 is arithmetical, being deﬁned in N by a formula ∃ x ∀ y ϕ, where ϕ is numeralwise determined by A E . Conversely, any arithmetical relation is deﬁned in N by some prenex formula. The quantiﬁer-free part of this prenex formula deﬁnes a recursive relation (since quantiﬁer-free formulas are numeralwise determined by A E ). Consequently, the deﬁned relation falls somewhere in the hierarchy. The technique of “collapsing” ∃∃ · · · ∃ quantiﬁers used in the proof of Theorem 35E (and its dual technique for ∀∀ · · · ∀) can be used to good advantage here.

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Figure 10. Picture of PN.

Thus we have the following result, which relates deﬁnability in N to the hierarchy we have just built up from the recursive relations: THEOREM 35K A relation on the natural numbers is arithmetical (i.e., deﬁnable in N) iff it is in k for some k, and this property in turn is equivalent to being in l for some l. In particular, any recursively enumerable relation is arithmetical, as noted previously. There are certain tricks which are useful in locating speciﬁc arithmetical relations in the hierarchy. For example, let A be the set of G¨odel numbers of formulas α such that for some n, A E ! α(Sn 0)

and

(∀i < n) A E ! ¬ α(Si 0).

Then a ∈ A iff [a is the G¨odel of a wff α] and ∃n∃D[D is a deduction from A E of α(Sn 0)] and (∀i < n)(∃Di )[Di is a deduction from A E of ¬ α(Si 0)]. The parts in square brackets are recursive so we count the remaining quantiﬁers. The bounded quantiﬁer “∀i < n” need not be counted. For we have (∀i < n)(∃d) d, i ∈ P ⇔ (∃d)(∀i < n) (d)i , i ∈ P. Use of this fact lets us push the bounded quantiﬁer inward until it merges with the recursive part. Consequently, A ∈ 1 . The following theorem generalizes Theorem 35I.

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THEOREM 35L Let A be a set of sentences such that A is in k , where k > 0. Then Cn A is also in k . PROOF. Return to the proofs of items 18 and 20 of Section 3.4. We had there: a ∈ Cn A ⇔ a is the G¨odel number of a sentence and ∃d[d is a sequence number and the last component of d is a and for every i less than lh d, either (1) (d)i ∈ A, (2) (d)i is the G¨odel number of a logical axiom, or (3) for some j and l less than i, (d) j = h(() ∗ (d)l ∗ h(→) ∗(d)i ∗ h())]. Since A ∈ k in (1) we must replace “(d)i ∈ A” by something of the form ∈Q ∃b1 ∀b2 · · · bk (d)i , b for recursive Q. It remains to convert the result into an English prenex expression in k form. We suggest that the reader set k = 2 and write out this expression; the device used in the preceding example will help.

Exercises 1. Show that there is no recursive set R such that Cn A E ⊆ R and {σ | ( ¬ σ ) ∈ Cn A E } ⊆ R, the complement of R. (This result can be stated: The theorems of A E cannot be recursively separated from the refutable sentences.) Suggestion: Make a sentence σ saying “My G¨odel number is not in R.” Look to see where σ is. 2. Let A be a recursive set of sentences in a recursively numbered language with 0 and S. Assume that every recursive relation is representable in the theory Cn A. Further assume that A is ω-consistent; i.e., there is no formula ϕ such that A ! ∃ x ϕ(x) and for all a ∈ N, A ! ¬ ϕ(Sa 0). Construct a sentence σ indirectly asserting that it is not a theorem of A, and show that neither A ! σ nor A ! ¬ σ . Suggestion: See Section 3.0. Remark: This is a version of the incompleteness theorem that is closer to G¨odel’s original 1931 argument. Note that there is no requirement that the axioms A be true in N. Nor is it required that A include A E ; the ﬁxed-point argument can still be applied. 3. Let T be a theory in a recursively numbered language (with 0 and S). Assume that all recursive subsets of N are weakly representable in T . Show that T is not recursive. Suggestion: Construct a binary relation P such that any weakly representable subset of N equals {b | a, b ∈ P} for some a, and such that P is recursive if T is.

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A Mathematical Introduction to Logic Consider the set H = {b | b, b ∈ / P}. See Section 3.0. The argument given there for the “diagonal approach” in the special case T = Th N can be adapted here. Remark: This exercise gives a version of the result, “Any sufﬁciently strong theory is undecidable.” 4. Show that there exist 2ℵ0 nonisomorphic countable models of Th N. Suggestion: For each set A of primes, make a model having an element divisible by exactly the primes in A. 5. (Lindenbaum) Let T be a decidable consistent theory (in a reasonable language). Show that T can be extended to a complete decidable consistent theory T . Suggestion: Examine in turn each sentence σ ; add either σ or ¬ σ to T . But take care to maintain decidability. 6. Consider the language of equality, having ∀ as its only parameter. Let λn be the translation of “There are at least n things,” cf. the proof of Theorem 26A. Call a formula simple iff it can be built up from atomic formulas and the λn ’s by use of connective symbols (but no quantiﬁers). Show how, given any formula in the language of equality, we can ﬁnd a logically equivalent simple formula. Suggestion: View this as an elimination-of-quantiﬁers result (where the quantiﬁers in λn do not count). Use Theorem 31F. 7. (a) Assume that A and B are subsets of N belonging to k (or k ). Show that A ∪ B and A ∩ B also belong to k (or k , respectively). (b) Assume that A is in k (or k ) and that the functions f 1 , . . . , f m are recursive. Show that { a : f 1 ( a ), . . . , f m ( a ) ∈ A} is also in k (or k , respectively). Suggestion: First do this for 1 . Then observe that the argument used can be generalized. 8. Let T be a theory in a recursively numbered language (with 0 and S). Let n be ﬁxed, n ≥ 0. Assume that all subsets of N in n are weakly representable in T . Show that T is not in n . (Observe that Exercise 3 is a special case of this, wherein n = 0. The suggestions given there carry over to the present case.) 9. Show that {σ | A E ; σ is ω-consistent} (see Exercise 2) is a 3 set. 10. The theory Cn A E has many complete extensions, of which Th N is but one. How many? That is, what is the cardinality of the set of complete theories (in the language) that extend A E ?

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SECTION 3.6 Recursive Functions We have used recursive functions (i.e., functions that, when viewed as relations, are recursive) to obtain theorems of incompleteness and undecidability of theories. But the class of recursive functions is also an interesting class in its own right, and in this section we will indicate a few of its properties. Recall that by Church’s thesis, a function is recursive iff it is computable by an effective procedure (page 210). This fact is responsible for much of the interest in recursive functions. At the same time, this fact makes possible an intuitive understanding of recursiveness, which greatly facilitates the study of the subject. Suppose, for example, that you are suddenly asked whether or not the inverse of a recursive permutation of N is recursive. Before trying to prove this, you should ﬁrst ask yourself the intuitive counterpart: Is the inverse of a computable permutation f also computable? You then — one hopes — perceive that the answer is afﬁrmative. To compute f −1 (3), you can compute f (0), f (1), . . . until for some k it is found that f (k) = 3. Then f −1 (3) = k. Having done this, you have gained two advantages. For one, you feel certain that the answer to the question regarding recursive permutations must also be afﬁrmative. And secondly, you have a good outline of how to prove this; the proof is found by making rigorous the intuitive proof. This strategy for approaching problems involving recursiveness will be very useful in this section. Before proceeding, it might be wise to summarize here some of the facts about recursive functions we have already established. We know that a function f is recursive iff it (as a relation) is representable in Cn A E , by Theorem 34A. Consequently, every recursive function is weakly representable in this theory. In Section 3.3 a repertoire of recursive functions was developed. In addition, it was shown that the class of recursive functions is closed under certain operations, such as composition (Theorem 33L) and the “least-zero” operator (Theorem 33M). We also know of a few functions that are not recursive. There are uncountably many (to be exact, 2ℵ0 ) functions from Nm into N altogether, but only countably many of them can be recursive. So an abundance of nonrecursive functions exists, despite the fact that the most commonly met functions (such as the polynomials) were shown in Section 3.3 to be recursive. By catalog item 1 of Section 3.3, the characteristic function of a nonrecursive set is nonrecursive. For example, if f (a) = 1 whenever a is the G¨odel number of a member of Cn A E and f (a) = 0 otherwise, then f is not recursive.

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Normal Form For any computable function, such as the polynomial function a 2 +3a + 5, one can in principle design a digital computer into which one feeds a and out of which comes a 2 + 3a + 5 (Fig. 11). But if you then want a different function, you must build a different computer. (Or change the wiring in the one you have.) It was recognized long ago that it is usually more desirable to build a single general-purpose stored-program computer. Into this you feed both a and the program for computing your polynomial (Fig. 12). This “universal” computer requires two inputs, and it will compute any one-place computable function (if supplied with enough memory space), provided that the right program is fed into it. Of course, there are some programs that do not produce any function on N, as many a programmer has, to his sorrow, discovered. (Such programs produce malfunctions instead!)

Figure 11. Special-purpose computer.

Figure 12. General-purpose computer.

In this subsection and the next, we will repeat what has just been said, but with recursive functions and with proofs. For our universal computer we will have a recursive relation T1 and a recursive function U . Then for any recursive f : N → N there will exist an e (analogous to the program) such that f (a) = U (the least k such that e, a, k ∈ T1 ) = U (μk e, a, k ∈ T1 ), where the second equation is to be understood as being an abbreviation for the ﬁrst. Actually e will here be the G¨odel number of a formula

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ϕ that represents (or at least weakly represents) f in Cn A E . And the numbers k for which e, a, k ∈ T1 will encode both f (a) and G of a deduction from A E of ϕ(Sa 0, S f (a) 0). DEFINITION. For each positive integer m, let Tm be the (m + 2)-ary relation to which an (m + 2)-tuple e, a1 , . . . , am , k belongs iff (i) e is the G¨odel number of a formula ϕ in which only v1 , . . . , vm , vm+1 occur free; (ii) k is a sequence number of length 2, and (k)0 is G of a deduction from A E of ϕ(Sa1 0, . . . , Sam 0, S(k)1 0). The idea here is that for any one-place recursive function f we can ﬁrst of all take e to be the G¨odel number of a formula ϕ weakly representing f (as a relation). Then we know that for any a and b, A E ! ϕ(Sa 0, Sb 0)

iff

b = f (a).

So any number k meeting clause (ii) of the deﬁnition must equal

(k)0 , f (a), where (k)0 is G of a deduction of ϕ(Sa 0, S f (a) 0) from A E . (We have departed from the usual deﬁnition of Tm here by not requiring that k be as small as possible.) Take for the “upshot” function U the function U (k) = (k)1 . This U is recursive and in the situation described in the preceding paragraph we have U (k) = f (a). LEMMA 36A For each m, the relation Tm is recursive. PROOF, FOR m = 2. e, a1 , a2 , k ∈ T2 iff e is the G¨odel number of a formula, (∀ v1 ∀ v2 ∀ v3 ) ∗ e is the G¨odel number of a sentence, k is a sequence number of length 2, and (k)0 is G of a deduction from A E of Sb(Sb(Sb(e, v1 , g(a1 )), v2 , g(a2 )), v3 , g((k)1 )), where g(n) = Sn 0. From Section 3.4 we know all this to be recursive. THEOREM 36B (a) For any recursive function f : Nm → N, there is an e such that for all a1 , . . . , am , f (a1 , . . . , am ) = U (μk e, a1 , . . . , am , k ∈ Tm ). (In particular, such a number k exists.) (b) Conversely, for any e such that ∀a1 · · · am ∃k e, a1 , . . . , am , k ∈ Tm , the function whose value at a1 , . . . , am is U (μk e, a1 , . . . , am , k ∈ Tm ) is recursive.

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A Mathematical Introduction to Logic PROOF. Part (b) follows immediately from the fact that U and Tm are recursive. As for part (a), we take for e the G¨odel number of a formula ϕ weakly representing f in Cn A E . Given any a , we know that A E ! ϕ(Sa1 0, . . . , Sam 0, S f (a ) 0). If we let d be G of a a ) ∈ Tm . deduction from A E of this sentence, then e, a , d, f ( Hence there is some k for which e, a , k ∈ Tm . And for any such k, we know that A E ! ϕ(Sa1 0, . . . , Sam 0, S(k)1 0), since (k)0 is G a ) by our choice of a deduction. Consequently, U (k) = (k)1 = f ( of ϕ. Thus we have U (μk e, a , k ∈ Tm ) = f ( a ). This theorem, due to Kleene in 1936, shows that every recursive function is representable in the normal form f ( a ) = U (μk e, a , k ∈ Tm ). Thus a computing machine able to calculate U and the characteristic function of T1 is a “universal” computer for one-place recursive functions. The input e corresponds to the program, and it must be chosen with care if any output is to result (i.e., if there is to be any k such that e, a, k ∈ T1 ).

Recursive Partial Functions The theory of recursive functions becomes more natural if we consider the broader context of partial functions. DEFINITION. An m-place partial function is a function f with / dom f , then f ( a ) is said to dom f ⊆ Nm and ran f ⊆ N. If a ∈ be undeﬁned. If dom f = Nm , then f is said to be total. The reader is hereby cautioned against reading too much into our choice of the words “partial” and “total” (or the word “undeﬁned,” for that matter). A partial function f may or may not be total; the words “partial” and “total” are not antonyms. We will begin by looking at those partial functions that are informally computable.

DEFINITION. An m-place partial function f is computable iff there is an effective procedure such that (a) given an m-tuple a in dom f , the procedure produces f ( a ); and (b) given an m-tuple a not in dom f , the procedure produces no output at all.

This deﬁnition extends the one previously given for total functions. At that time we proved a result (Theorem 33H), part of which generalizes to partial functions.

THEOREM 36C An m-place partial function f is computable iff f (as an (m + 1)-ary relation) is effectively enumerable.

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PROOF. The proof is reminiscent of the proof of another result, Theorem 17E. First suppose we have a way of effectively enumerating f . Given an m-tuple a , we examine the listing of the relation as the procedure churns it out. If and when an (m + 1)tuple beginning with a appears, we print out its last component as f ( a ). Conversely, assume that f is computable, and ﬁrst suppose that f is a one-place partial function. We can enumerate f as a relation by the following procedure: 1. Spend one minute calculating f (0). 2. Spend two minutes calculating f (0), then two minutes calculating f (1). 3. Spend three minutes calculating f (0), three minutes calculating f (1), and three minutes calculating f (2). And so forth. Of course, whenever one of these calculations produces any output, we place the corresponding pair on the list of members of the relation f . For a computable m-place partial function, instead of calculating the value of f at 0, 1, 2, . . . we calculate its value at (0)0 , . . . , (0)m−1 , (1)0 , . . . , (1)m−1 , (2)0 , . . . , (2)m−1 , etc. In the case of a computable total function f , we were also able to conclude that f was a decidable relation. But this may fail for a nontotal f . For example, let 0 if a ∈ Cn A E , f (a) = undeﬁned otherwise. Then f is computable. (We compute f (a) by enumerating Cn A E and looking for a.) But f is not a decidable relation, lest Cn A E be decidable. On the basis of this example and the foregoing theorem, we select our deﬁnition for the precise counterpart of the concept of computable partial function. DEFINITION. A recursive partial function is a partial function that, as a relation, is recursively enumerable. The reader should be warned that “recursive partial function” is an indivisible phrase; a recursive partial function need not be (as relation) recursive. But at least for a total function our terminology is consistent with past practice. THEOREM 36D Let f : Nm → N be a total function. Then f is a recursive partial function iff f is recursive (as a relation). PROOF. If f is recursive (as a relation), then a fortiori f is recursively enumerable. Conversely, suppose that f is recursively

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A Mathematical Introduction to Logic enumerable. Since f is total, f ( a) =

b ⇐⇒ ∃c[ f ( a) = c & b =

c]. The form of the right-hand side shows that the complement of f is also recursively enumerable. Thus by Theorem 35F, f is recursive. In ﬁrst discussing normal form results, we pictured a two-input device (Fig. 13). For any computable partial function, there is some program that computes it. But now the converse holds: Any program will produce some computable partial function. Of course many programs will produce the empty function, but that is a computable partial function.

Figure 13. Computer with program for f .

For the recursive partial functions the same considerations apply. Deﬁne, for each e ∈ N, the m-place partial function [[e]]m by [[e]]m (a1 , . . . , am ) = U (μk e, a1 , . . . , am , k ∈ Tm ). The right-hand side is to be understood as undeﬁned if there is no such k. In other words, a ∈ dom[[e]]m

iff

∃k e, a1 , . . . , am , k ∈ Tm ,

in which case the value [[e]]m ( a ) is given by the above equation. The following theorem is an improved version of Theorem 36B: NORMAL FORM THEOREM (KLEENE, 1943) (a) The (m + 1)-place partial function whose value at e, a1 , . . . , am is [[e]]m (a1 , . . . , am ) is a recursive partial function. (b) For each e ≥ 0, [[e]]m is an m-place recursive partial function. (c) Any m-place recursive partial function equals [[e]]m for some e. PROOF.

(a) We have

[[e]]m ( a ) = b ⇔ ∃k[ e, a , k ∈ Tm & U(k) = b & (∀k < k) e, a ,k ∈ / Tm ].

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The part in square brackets is recursive, so the function (as a relation) is recursively enumerable. (b) The above proof still applies, e now being held ﬁxed. (c) Let f be an m-place recursive partial function, so that { a , b | f ( a ) = b} is recursively enumerable. Hence there is a formula ϕ that weakly represents this relation in Cn A E . We claim a ) = b, then A E ! ϕ(Sa1 0, . . . , Sam 0, that f = [[ϕ]]m . For if f ( b S 0). Hence there is a k such that ϕ, a , k ∈ Tm . For any such

b. Simk, U (k) = b, since A E ϕ(Sa1 0, . . . , Sam 0, Sc 0) for c = ilarly, if f ( a ) is undeﬁned, then A E ϕ(Sa1 0, . . . , Sam 0, Sc 0) for any c, whence [[ϕ]]m is undeﬁned here also. Part (a) of the normal form theorem (in the case m = 1) tells us that the function deﬁned by the equation (e, a) = [[e]]1 (a) = U (μk e, a, k ∈ T1 ) is a recursive partial function. And part (c) tells us that is “universal” in the sense that we can get any one-place recursive partial function from by holding the ﬁrst variable ﬁxed at a suitable value. The informal counterpart of the universal function is the computer operating system. The operating system takes two inputs, the program e and the data a. And it runs the program on that data. But the operating system itself is computable as a two-place partial function. The proof of normal form theorem gives us a way to compute the values of our “operating system” , albeit in an extremely inefﬁcient way. The straightforward idea of “looking at the program e and doing what it says to the data a” has been obscured, to say the least. The function [[e]]m is said to be the m-place recursive partial function with index e. Part (c) of the normal form theorem tells us that every recursive partial function has an index. The proof shows that the G¨odel number of a formula weakly representing a function is always an index of the function. We now have a convenient indexing [[0]]1 , [[1]]1 , . . . of all the oneplace recursive partial functions. Function [[e]]1 is produced by the “instructions” encoded by e. Of course, that function will be empty unless e is the G¨odel number of a formula and certain other conditions are met. All the recursive total functions are included in our enumeration of recursive partial functions. But we cannot tell effectively by looking at a number e whether or not it is the index of a total function: THEOREM 36E PROOF.

{e | [[e]]1 is total} is not recursive.

Call this set A. Consider the function deﬁned by [[a]]1 (a) + 1 if a ∈ A, f (a) = 0 if a ∈ / A.

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A Mathematical Introduction to Logic Then f , by its construction, is total. Is it recursive? We have f (a) = b ⇐⇒ [(a ∈ / A & b = 0) or (a ∈ A & ∃k( a, a, k ∈ T1 & b = U (k) + 1 & (∀ j < k) a, a, j ∈ / T1 ))]. Thus if A is recursive, then f (as a relation) is recursively enumerable. But then f is a total recursive function, and so equals [[e]]1 for some e ∈ A. But f (e) = [[e]]1 (e) + 1, so we cannot have f = [[e]]1 . This contradiction shows that A cannot be recursive. It is not hard to show that A is in 2 . This classiﬁcation is the best possible, as it can be shown that A is not in 2 . THEOREM 36F The set K = {a | [[a]]1 (a) is deﬁned} is recursively enumerable but not recursive. PROOF. K is recursively enumerable, since a ∈ K ⇔ ∃ k a, a, k ∈ T1 . To see that K cannot be recursive, consider the function deﬁned by [[a]]1 (a) + 1 if a ∈ K , g(a) = 0 if a ∈ / K. This is a total function. Exactly as in the preceding theorem, we have that K cannot be recursive. COROLLARY 36G (UNSOLVABILITY OF THE HALTING PROBLEM) lation

The re-

{ e, a | [[e]]1 (a) is deﬁned} is not recursive. PROOF. We have a ∈ K iff the pair a, a belongs to this relation. (Thus the problem of membership in K is “reducible” to the halting problem.) If this relation were recursive, then K would be, which is not the case. This corollary tells us that there is no effective way to tell, given a program e for a recursive partial function and an input a, whether or not the function [[e]]1 is deﬁned at a. We can obtain an indexing of the recursively enumerable relations by using the following characterization. THEOREM 36H A relation on N is recursively enumerable iff it is the domain of some recursive partial function. PROOF. The domain of any recursively enumerable relation is also recursively enumerable; cf. part 4 of Theorem 35E. In particular,

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the domain of any recursive partial function is recursively enumerable. Conversely, let Q be any recursively enumerable relation, where a = Q ⇔ ∃b a , b ∈ R with R recursive. Let f ( a ) = μb a , b ∈ R; i.e., f ( a ) = b ⇐⇒ a , b ∈ R & (∀c < b) a , c ∈ / R. Then f , as a relation, is recursive. Hence f is a recursive partial function. Clearly its domain is Q. Thus our indexing of the recursive partial functions induces an indexing of the recursively enumerable relations. Deﬁne We = dom[[e]]1 . Then W0 , W1 , W2 , . . . is a list of all recursively enumerable subsets of N. In Theorem 36E we showed that {e | We = N} is not recursive. Similarly, Theorem 36F asserts that {e | e ∈ We } is not recursive. Deﬁne a relation Q by Q = { e, a | a ∈ We }. Then Q is recursively enumerable, since e, a ∈ Q ⇔ ∃ k e, a, k ∈ T1 . Furthermore, Q is universal for recursively enumerable sets, in the sense that for any recursively enumerable A ⊆ N there is some e such that A = {a | e, a ∈ Q}. The unsolvability of the halting problem can be stated: Q is not recursive. We can apply the classical diagonal argument to “diagonalize out” of the list W0 , W1 , W2 , . . . of recursively enumerable sets. The set {a | a ∈ / Wa } cannot coincide with any Wq . In fact this set is exactly K , the complement of the set K in Theorem 36F. Because q ∈ K ⇐⇒ q ∈ / Wq , the set K cannot equal any Wq ; the number q witnesses the inequality of the two sets K and Wq . And there is more: Whenever Wq is a recursively enumerable subset of K , that is, Wq ⊆ K , then we can produce a number in K that is not in Wq . Such a number is q itself. To see this, observe that in the line displayed in the preceding paragraph we cannot have both sides false (q ∈ K and q ∈ Wq ) because Wq ⊆ K . So both sides are true. Theorem 36F asserts that K , although recursively enumerable, is not recursive. To show non-recursiveness, it sufﬁces to show that its complement K is not recursively enumerable. The preceding paragraph

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A Mathematical Introduction to Logic does this in a particularly strong way, thereby giving us a second proof of Theorem 36F. At this point, let us reconsider the G¨odel incompleteness theorem, from the computability point of view. The set K is recursively enumerable (i.e., 1 ). It follows (cf. Theorem 35K) that K is arithmetical; that is, K is deﬁnable in the structure N. So there is a formula (v1 ) with just v1 free that deﬁnes K in N. And so the set K is deﬁned in N by the formula ¬ (v1 ). Thus we have a ∈ K ⇐⇒ ( ¬ (Sa 0)) ∈ Th N. This fact tells how we can “reduce” questions about membership in the set K to questions about Th N. Imagine that we are given a number a, and we want to know whether or not a ∈ K . We can compute the number ( ¬ (Sa 0)). (Informally, it is clear that we can effectively compute this number. Formally, we apply item 5 from Section 3.4 to make sure we can recursively compute the number.) If we somehow had an oracle for Th N (i.e., a magic device that, given a number, would tell us whether or not that number was in Th N), then we could answer the question “Is a ∈ K ?” Now let us eliminate the magic. For sets A and B of natural numbers, we say that A is many-one reducible to B (in symbols, A ≤m B) iff there exists a total recursive function f such that for every number a, a ∈ A ⇐⇒ f (a) ∈ B. The earlier example tells us that K ≤m Th N. More generally, the argument shows that any arithmetical set is many-one reducible to Th N. LEMMA 36I Assume that A and B are sets of natural numbers with A ≤m B. (a) If B is recursive, then A is also recursive. (b) If B is recursively enumerable, the A is also recursively enumerable. (c) If B is n for some n, the A is also n for that n. PROOF. Part (a) is already familiar; it was, in different terminology, catalog item 2 in Section 3.3. Part (b) is essentially part (a) “plus a quantiﬁer.” That is, because B is recursively enumerable, we know that for some recursive binary relation Q, c ∈ B ⇐⇒ ∃b Q(c, b). If f is the total recursive function that many-one reduces A to B, then every number a, a ∈ A ⇐⇒ f (a) ∈ B ⇐⇒ ∃b[Q( f (a), b)].

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The part in square brackets is recursive (i.e., { a, b | Q( f (a), b)} is recursive), as in part (a) of the lemma. So we have A in the required form to be recursively enumerable. Part (c) is essentially part (a) “plus n quantiﬁers” and is proved like part (b). Our reason for examining the particular set K is that it gives us the following consequence: INCOMPLETENESS THEOREM Th N is not recursively axiomatizGODEL ¨ able. PROOF. Th N cannot be recursively enumerable, lest K be recursively enumerable, by the preceding lemma. But any recursively axiomatizable theory would be recursively enumerable (item 20 of Section 3.4; also Theorem 35I). In starkest terms, the situation is this: Any recursively axiomatizable theory is recursively enumerable. But Th N is not recursively enumerable. So any recursively axiomatizable subtheory must be incomplete. It will be worth while to go over this proof again, but replacing negative statements (such-and-such a set does not have a particular property) by positive statements. Assume that T is any recursively axiomatizable subtheory of Th N. (So by the above theorem, T is incomplete.) We want to lay our hands on a sentence demonstrating the incompleteness. We have made a total recursive function f that many-one reduces K to Th N, namely f (a) = ( ¬ (Sa 0)); then for every a, a ∈ K ⇐⇒ f (a) ∈ Th N. And f (a) is (the G¨odel number of ) the sentence saying “a ∈ / K .” Consider the set J of numbers deﬁned by the condition a ∈ J ⇐⇒ f (a) ∈ T. Thus J is the set of numbers that T “knows” are not in K . There are two observations to be made concerning J : First, J is recursively enumerable. It is many-one reduced by f to the recursively enumerable set T ; apply Lemma 36I(b). / K, Secondly, J ⊆ K . We have T ⊆ Th N, so if T knows that a ∈ then really a ∈ / K: a ∈ J ⇐⇒ f (a) ∈ T '⇒ f (a) ∈ Th N ⇐⇒ a ∈ K . So J is a recursively enumerable subset of K . It is a proper subset, because K is not recursively enumerable. That is, there is some number q with q ∈ K and q ∈ / J . Consequently, f (q) ∈ Th N but f (q) ∈ / T . That is, the sentence ( ¬ (Sq 0)) is true (in N) but fails to be in T , thereby demonstrating the incompleteness of T .

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A Mathematical Introduction to Logic And what does this sentence “say”? For q, we can take any number / J. for which Wq = J . Then q ∈ K and q ∈ Here then is the situation: ( ¬ (Sq 0))

says q ∈ /K i.e., q ∈ / Wq i.e., q ∈ / J since Wq = J i.e., f (q) ∈ / T by deﬁnition of J i.e., T ! ( ¬ (Sq 0)).

The sentence we made to witness the incompleteness of T asserts its own unprovability in the axiomatizable theory T ! The computability approach and the self-reference approach to G¨odel’s incompleteness theorem are not so different after all. Moreover, the computability approach is close to the diagonalization approach (of Section 3.0), but with the diagonal argument moved to a different context.

Reduction of Decision Problems1 Suppose we have a two-place recursive partial function f . Then we claim that, for example, the function g deﬁned by g(a) = f (3, a) is also a recursive partial function. On the basis of informal computability this is clear; one computes g by plugging in 3 for the ﬁrst variable and then following the instructions for f . A proof can be found by formalizing this argument. There is some formula ϕ = ϕ(v1 , v2 , v3 ) that weakly represents f (as a relation) in Cn A E . Then g is weakly represented by ϕ(S3 0, v1 , v2 ), provided that v1 and v2 are substitutable in ϕ for v2 and v3 . (If not, we can always use an alphabetic variant of ϕ.) Now all this is not very deep. But by standing back and looking at what was said, we perceive a more subtle fact. We were able to transform effectively the instructions for f into instructions for g. So there should be a recursive function that, given an index for f and the number 3, will produce an index for g. The following formulation of this fact is sometimes known by the cryptic name of “the S-m-n theorem.” PARAMETER THEOREM For each m ≥ 1 and n ≥ 1 there is a recursive function ρ such that for any e, a , b, [[e]]m+n (a1 , . . . , am , b1 , . . . , bn ) = [[ρ(e, a1 , . . . , am )]]n (b1 , . . . , bn ). (Equality here means of course that if one side is deﬁned, then so also is the other side, and the values coincide. Sometimes a special symbol “#” is used for this role.)

1

The remainder of this section can be skipped on a ﬁrst reading.

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On the left side of the equation a consists of arguments for the function [[e]]m+n ; on the right side a consists of parameters upon which the function [[ρ(e, a )]]n depends. In the example we had m = n = 1 and a1 = 3. Since ρ depends on m and n, the notation “ρnm ” would be logically preferable. But, in fact, we will use simply “ρ.” PROOF, FOR m = n = 1. It is possible to give a proof along the lines indicated by the discussion that preceded the theorem. But to avoid having to cope with alphabetic variants, we will adopt a slightly different strategy. We know from the normal form theorem that the three-place partial function h deﬁned by h(e, a, b) = [[e]]2 (a, b) is a recursive partial function. Hence there is a formula ψ that weakly represents h (as a relation). We may suppose that in ψ the variables v1 and v2 are not quantiﬁed. We can then take ρ(e, a) = ψ(Se 0, Sa 0, v1 , v2 ) = Sb(Sb(Sb(Sb(ψ, v1 , Se 0), v2 , Sa 0), v3 , v1 ), v4 , v2 ). Then ρ(e, a) is the G¨odel number of a formula weakly representing the function g(b) = [[e]]2 (a, b). Hence it is an index of g. We will utilize the parameter theorem to show that certain sets are not recursive. We already know that K = {a | [[a]]1 (a) is deﬁned} is not recursive. For a given nonrecursive set A we can sometimes ﬁnd a (total) recursive function g such that a ∈ K ⇔ g(a) ∈ A or a (total) recursive function g such that a∈ / K ⇔ g (a) ∈ A. In either case it then follows at once that A cannot be recursive lest K be. In the former case we have K ≤m A and A is not 1 (by Lemma 36I); in the latter case K ≤m A and A is not 1 . In either case, A is not recursive. The function g or g can often be obtained from the parameter theorem. EXAMPLE.

{a | Wa = ∅} is not recursive.

PROOF. Call this set A. First, note that A ∈ 1 , since Wa = ∅ iff ∀b∀k a, b, k ∈ / T1 . Consequently, K cannot be many-one reducible to A, but it is reasonable to hope that K might be. That is, we want a total recursive function g such that [[a]]1 (a) is undeﬁned ⇔ dom[[g(a)]]1 = ∅.

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A Mathematical Introduction to Logic This will hold if for all b, [[g(a)]]1 (b) = [[a]]1 (a). So start with the recursive partial function f (a, b) = [[a]]1 (a) and let g(a) = ρ( fˆ , a), where fˆ is an index for f . Then [[g(a)]]1 (b) = [[ρ( fˆ , a)]]1 (b) = f (a, b) = [[a]]1 (a). Thus this g shows that K is many-one reducible to A.

THEOREM 36J (RICE, 1953) Let C be a set of one-place recursive partial functions. Then the set {e | [[e]]1 ∈ C} of indices of members of C is recursive iff either C is empty or C contains all one-place recursive partial functions. PROOF. Only one direction requires proof. Let IC = {e | [[e]]1 ∈ C} be the set of indices of members of C. Case I: The empty function ∅ is not in C. If nothing at all is in C we are done, but suppose some function ψ is in C. We can show that K is many-one reducible to IC if we have a recursive total function g such that ψ if a ∈ K , [[g(a)]]1 = ∅ if a ∈ / K. For then a ∈ K ⇔ [[g(a)]]1 ∈ C ⇔ g(a) ∈ IC . We can obtain g from the parameter theorem by deﬁning g(a) = ρ(e, a), where

[[e]]2 (a, b) =

ψ(b) undeﬁned

if a ∈ K , if a ∈ / K.

The above is a recursive partial function, since [[e]]2 (a, b) = c ⇔ a ∈ K & ψ(b) = c and the right-hand side is recursively enumerable. Case II: ∅ ∈ C. Then apply case I to the complement C of C. We can then conclude that IC is not recursive. But IC is the complement of IC , so IC cannot be recursive. Thus in either case, IC is not recursive. EXAMPLES. For any ﬁxed e, the set {a | Wa = We } is not recursive, as a consequence of Rice’s theorem. In particular, {a | Wa = ∅} is not recursive, a result proved in an earlier example. For two other applications of Rice’s theorem, we can say that {a | Wa is inﬁnite} and {a | Wa is recursive} are not recursive.

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Register Machines There are many equivalent deﬁnitions of the class of recursive functions. Several of these deﬁnitions employ idealized computing devices. These computing devices are like digital computers but are free of any limitation on memory space. The ﬁrst deﬁnition of this type was published by Alan Turing in 1936; similar work was done by Emil Post at roughly the same time. We will give here a variation on this theme, due to Shepherdson and Sturgis (1963). A register machine will have a ﬁnite number of registers, numbered 1, 2, . . . , K . Each register is capable of storing a natural number of any magnitude. The operation of the machine will be determined by a program. A program is a ﬁnite sequence of instructions, drawn from the following list: I r (where 1 ≤ r ≤ K ). “Increment r .” The effect of this instruction is to increase the contents of register r by 1. The machine then proceeds to the next instruction in the program. D r (where 1 ≤ r ≤ K ). “Decrement r .” The effect of this instruction depends on the contents of register r . If that number is nonzero, it is decreased by 1 and the machine then proceeds, not to the next instruction, but to the following one. But if the number in register r is zero, the machine just proceeds to the next instruction. In summary: The machine tries to decrement register r and skips an instruction if it is successful. T q (where q is an integer–positive, negative, or zero). “Transfer q.” All registers are left unchanged. The machine takes as its next instruction the qth instruction following this one in the program (if q ≥ 0), or the |q|th instruction preceding this one (if q < 0). The machine halts if there is no such instruction in the program. An instruction of T 0 results in a loop, with the machine executing this one instruction over and over again. EXAMPLES 1. Program to clear register 7. Try to decrement 7. Go back and repeat. Halt. 2. Program to move a number from register r to register s. Clear register s

(Use the program of the ﬁrst example.) Take 1 from r . Halt when zero. Add 1 to s. Repeat.

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A Mathematical Introduction to Logic This program has seven instructions altogether. It leaves a zero in register r . 3. Program to add register 1 to registers 2 and 3.

4. (Addition) Say that a and b are in registers 1 and 2. We want a + b in register 3, and we want to leave a and b still in registers 1 and 2 at the end. Register contents Clear register 3. a b 0 Move number from register 1 to register 4. 0 b 0 a Add register 4 to registers 1 and 3. a b a 0 Move number from register 2 to register 4. a 0 a b Add register 4 to registers 2 and 3. a b a+b 0 This program has 27 instructions as it is written, but three of them are unnecessary. (In the fourth line we begin by clearing register 4, which is already clear.) At the end we have the number a back in register 1. But during the program register 1 must be cleared; this is the only way of determining the number a. 5. (Subtraction) Let a −. b = max(a − b, 0). We leave this program to the reader (Exercise 11). Now suppose f is an n-place partial function on N. Possibly there will be a program P such that if we start a register machine (having all the registers to which P refers) with a1 , . . . , an in registers 1, . . . , n and apply program P, then the following conditions hold: (i) If f (a1 , . . . , an ) is deﬁned, then the calculation eventually terminates with f (a1 , . . . , an ) in register n + 1. Furthermore, the calculation terminates by seeking a ( p +1)st instruction, where p is the length of P. (ii) If f (a1 , . . . , an ) is undeﬁned, then the calculation never terminates. If there is such a program P, we say that P calculates f . THEOREM 36K Let f be a partial function. Then there is a program that calculates f iff f is a recursive partial function. Thus by using register machines we arrive at exactly the class of recursive partial functions, a class we originally deﬁned in terms of representability in consistent ﬁnitely axiomatizable theories. The fact

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that such different approaches produce the same class of partial functions is evidence that this is a signiﬁcant class. OUTLINE OF PROOF. To show that the functions calculable by register machines are recursive partial functions, one “arithmetizes calculations” in the same spirit as we arithmetized deductions in Section 3.4. That is, one assigns G¨odel numbers to programs and to sequences of memory conﬁgurations. One then veriﬁes that the relevant concepts, translated into numerical relations by the G¨odel numbering, are all recursive. (After going through this, one perceives that, from a sufﬁciently general point of view, deductions and calculations are really about the same sort of thing.) Conversely, to show that the recursive partial functions are calculable by register machines, one can work through Sections 3.3 and 3.4 again, but where functions were previously shown to be representable in Cn A E , they must now be shown to be calculable by register machines. This is not as hard as it might sound, since after the ﬁrst few pages, the proofs are all the same as the ones used before. There is a reason for this similarity. It can be shown that the class of all recursive functions is generated from a certain handful of recursive functions by the operation of composition (in the sense of Theorem 33L) and the “least-zero” operator (Theorem 33M). Much of the work in Sections 3.3 and 3.4 amounts to a veriﬁcation of this fact. Thus once one has shown that each function in this initial handful is calculable by a register machine and that the class of functions calculable by register machines is closed under composition and the least-zero operator, then the earlier work can be carried over, yielding the calculability of all recursive functions.

Exercises 1. Deﬁne functions f and g by f (n) = g(n) =

0 1

if Goldbach’s conjecture is true, otherwise;

1

if in the decimal expansion of π there is a run of at least n consecutive 7’s, otherwise.

⎧ ⎨0 ⎩

Is f recursive? Is g recursive? (Goldbach’s conjecture says that every even integer greater than 2 is the sum of two primes. The ﬁrst edition of this book used Fermat’s last theorem here.)

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A Mathematical Introduction to Logic 2. Deﬁne the “diagonal” function d(a) = [[a]]1 (a) + 1. (a) Show that d is a recursive partial function. (b) By part (a), we have d = [[e]]1 for a certain number e. So on the one hand, d(e) = [[e]]1 (e) and on the other hand d(e) = [[e]]1 (e)+1. Can we cancel, to conclude that 0 = 1? Suggestion: Use the special symbol “#” to mean that either both sides of the equation are undeﬁned, or both sides are deﬁned and equal. Restate the argument in this notation. 3. (a) Show that the range of any recursive partial function is recursively enumerable. (b) Show that the range of a strictly increasing (i.e., f (n) < f (n + 1)) total recursive function f is recursive. (c) Show that the range of a nondecreasing (i.e., f (n) ≤ f (n + 1)) total recursive function f is recursive. 4. (a) Let A be a nonempty recursively enumerable subset of N. Show that A is the range of some total recursive function. (b) Show that any inﬁnite recursively enumerable subset of N includes an inﬁnite recursive subset. 5. Show that every recursive partial function has inﬁnitely many indices. 6. Give an example of a function f and a number e such that for all a, f (a) = U (μk e, a, k ∈ T1 ) but e is not the G¨odel number of a formula weakly representing in Cn A E . 7. Show that the parameter theorem can be strengthened by requiring ρ to be one-to-one. 8. Recall that the union of two recursively enumerable sets is recursively enumerable (Exercise 7 of Section 3.5). Show that there is a total recursive function g such that Wg(a,b) = Wa ∪ Wb . 9. Show that {a | Wa has two or more members} is in 1 but not in 1 . 10. Show that there is no recursively enumerable set A such that {[[a]]1 | a ∈ A} equals the class of total recursive functions on N. 11. Give register machine programs that calculate the following functions: (a) Subtraction, a −. b = max(a − b, 0). (b) Multiplication, a · b. (c) max(a, b). 12. Assume that there is a register machine program that calculates the n-place partial function f . Show that given any positive

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integers r1 , . . . , rn (all distinct), p, and k, we can ﬁnd a program Q such that whenever we start a register machine (having all the registers to which Q refers) with a1 , . . . , an in registers r1 , . . . , rn and apply program Q, then (i) if f (a1 , . . . , an ) is deﬁned, then the calculation eventually terminates with f (a1 , . . . , an ) in register p, with the contents of registers 1, 2, . . . , k (except for register p) the same as their initial contents, and furthermore the calculation terminates by seeking a (q + 1)st instruction, where q is the length of Q; (ii) if f (a1 , . . . , an ) is undeﬁned, then the calculation never terminates. 13. Let g : Nn+1 → N be a (total) function that is calculated by some register machine program. Let f (a1 , . . . , an ) = μb[g(a1 , . . . , an , b) = 0], where the right-hand side is undeﬁned if no such b exists. Show that the partial function f can be calculated by some register machine program. 14. Show that the following sets have the given location in the arithmetical hierarchy. (In each case, the given location is the best possible, but we will not prove that fact.) (a) {e | [[e]]1 is total} is 2 . (b) {e | We is ﬁnite} is 2 . (c) {e | We is coﬁnite} is 3 . (d) {e | We is recursive} is 3 . 15. Let Tot = {e | [[e]]1 is total}. Clearly Tot ⊂ K . Show that there is no recursive set A with Tot ⊆ A ⊆ K . Remark: This result includes Theorems 36E and 36F; the proofs used there can be adapted here. 16. (a) Show that each 2 set of natural numbers is, for some number e, the set {a | ∀b∃c T2 (e, a, b, c)}. (b) Show that the set {a | not ∀b∃c T2 (a, a, b, c)} is 2 but not 2 . ∗ (c) Generalize parts (a) and (b) to show that for each n, there is a set that is n but not n . 17. Assume that A is a set of natural numbers that is arithmetical but is not m . Use the argument of page 256 to show that Th N is not m . Remark: Exercises 16 and 17 give a proof of Tarski’s theorem (that Th N is not arithmetical) from computability theory.

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SECTION 3.7 Second Incompleteness Theorem Let us return once again to item 20 in Section 3.4. Suppose that we have a recursively axiomatizable theory T given by a recursive set A of axioms (i.e., A is recursive). Then as in item 20 a ∈ T ⇐⇒ ∃d [d is the number of a deduction from A and the last component of d is a and a is the G¨odel number of a sentence]. The set of pairs a, d meeting the condition in brackets is recursive; let π(v1 , v2 ) be a formula — chosen in some natural way — that numeralwise represents that binary relation in A E . For any sentence σ , we can express “T ! σ ” by the sentence ∃ v2 π(Sσ 0, v2 ). Let us give that sentence a name; deﬁne PrbT σ = ∃ v2 π(Sσ 0, v2 ). (Here Prb abbreviates “provable.” The subscript should perhaps be “ A” instead of “T ”; in constructing the sentence we utilize the recursiveness of the set A of axioms.) LEMMA 37A

Let T be a recursively axiomatizable theory as above.

(a) Whenever T ! σ then A E ! PrbT σ . (b) If in addition T includes A E , then T has the “reﬂection” property: T ! σ '⇒ T ! PrbT σ . PROOF. If T ! σ then we can let d be the number of a deduction of σ from the axioms A for T . We have A E ! π(Sσ 0, Sd 0), and hence A E ! PrbT σ . This gives part (a), from which part (b) follows immediately. Thus under modest assumptions, whenever T proves a sentence, it knows that it proves the sentence. Note that part (b) does not say that T ! (σ → PrbT σ ). For example, if σ is true (in N) but unprovable from A E , then the sentence (σ → Prb A E σ ) is not provable from A E , and in fact is false in N. Returning now to the proof of the G¨odel incompleteness theorem (in the self-reference approach), we can apply the ﬁxed-point lemma to obtain a sentence σ asserting its own unprovability in T : A E ! (σ ↔ ¬ PrbT σ ). The following lemma provides part of the incompleteness theorem (the other part being Exercise 2 in Section 3.5):

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LEMMA 37B Let T be a recursively axiomatizable theory including A E and let σ be obtained from the ﬁxed-point lemma as above. If T is consistent, then T ! σ . PROOF T ! σ ⇒ T ! PrbT σ by reﬂection ⇒ T ! ¬σ by choice of σ

whence T is inconsistent.

So far, this lemma merely reﬂects ideas employed in Section 3.5, and the proof of Lemma 37B was not very complex. And that is exactly the point: The proof is not very complex, so perhaps it can be carried out within the theory T , if T is “sufﬁciently strong.” That, we can hope that the steps PrbT σ → PrbT PrbT σ → PrbT ¬ σ → PrbT 0 = S0 can be carried out in a sufﬁciently strong extension T of A E . If so, we get a remarkable conclusion. Let Cons T be the sentence ¬ PrbT 0 = S0, which we think of as saying “T is consistent.” (Here 0 = S0 is chosen simply as a convenient sentence refutable from A E .) If T lets us carry out the steps in the preceding paragraph, then we can conclude: T ! Cons T,

unless T is inconsistent

(Of course, an inconsistent theory contains every sentence, including sentences asserting — falsely — the theory’s consistency. The situation we are ﬁnding here is that, under suitable assumptions, this is the only way that a theory can prove its own consistency.) Let’s check the details: Suppose T ! Cons T . Then by the preceding paragraph, T ! ¬ PrbT σ . By choice of σ , we then get T ! σ . Lemma 37B then applies. To make matters less vague, call the theory T sufﬁciently strong if it meets the following three “derivability” conditions. 1. A E ⊆ T . This implies by Lemma 37A that T has the reﬂection property, T ! σ ⇒ T ! PrbT σ . 2. For any sentence σ , T ! ( PrbT σ → PrbT PrbT σ ). This is the reﬂection property, formalized within T . 3. For any sentences ρ and σ , T ! (PrbT (ρ → σ ) → (PrbT ρ → PrbT σ )). This is modus ponens, formalized within T . FORMALIZED LEMMA 37B Assume that T is a sufﬁciently strong recursively axiomatizable theory, and let σ be a sentence such that A E ! (σ ↔ ¬ PrbT σ ). Then T ! (Cons T → ¬ PrbT σ ).

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A Mathematical Introduction to Logic PROOF. We put the pieces together carefully. By the choice of σ we get T ! (σ → (PrbT σ → 0 = S0)). Applying ﬁrst reﬂection and then formalized modus ponens to this formula yields T ! (PrbT σ → PrbT (PrbT σ → 0 = S0)) after which another application of formalized modus ponens yields T ! (PrbT σ → (PrbT PrbT σ → ¬ Cons T )). The formula displayed above (to the right of the turnstile), together with PrbT σ → PrbT PrbT σ (formalized reﬂection) imply by sentential logic PrbT σ → ¬ Cons T . GODEL ¨ ’S SECOND INCOMPLETENESS THEOREM (1931) Assume that T is a sufﬁciently strong recursively axiomatizable theory. Then T ! Cons T if and only if T is inconsistent. PROOF. If T ! Cons T then by Formalized Lemma 37B we have T ! ¬ PrbT σ whence by our choice of σ , we have T ! σ . We conclude from the (unformalized) Lemma 37B that T is inconsistent. We can squeeze a bit more out of these ideas. Lemma 37B can be regarded as a special case (where τ is 0 = S0) of the following: LEMMA 37C Let T be a recursively axiomatizable theory including A E , let τ be a sentence, and let σ be obtained from the ﬁxed-point lemma so that A E ! (σ ↔ (PrbT σ → τ )). If T ! σ , then T ! τ . PROOF. We can think of σ as saying, “If I am provable, then τ .” If T ! σ then by reﬂection T ! PrbT σ . By the choice of σ , we have T ! τ . Actually we are not interested in this lemma, but in its formalization: FORMALIZED LEMMA 37C Assume that T is a sufﬁciently strong recursively axiomatizable theory. Let τ be a sentence, and let σ be a sentence such that A E ! (σ ↔ (PrbT σ → τ )). Then T ! ( PrbT σ → PrbT τ ).

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269

We proceed as before. By the choice of σ we get T ! (σ → (PrbT σ → τ )).

Applying ﬁrst reﬂection and then formalized modus ponens to this formula yields T ! (PrbT σ → PrbT (PrbT σ → τ )) after which another application of formalized modus ponens yields T ! (PrbT σ → (PrbT PrbT σ → PrbT τ )). The formula displayed above (to the right of the turnstile), together with PrbT σ → PrbT PrbT σ (formalized reﬂection) imply by sentential logic PrbT σ → PrbT τ . LOB ¨ ’S THEOREM (1955) Assume that T is a sufﬁciently strong recursively axiomatizable theory. If τ is any sentence for which T ! ( PrbT τ → τ ), then T ! τ . Clearly if T ! τ , then T ! (ρ → τ ) for any sentence ρ. So the conclusion to L¨ob’s theorem can be stated T ! (PrbT τ → τ ) ⇐⇒ T ! τ. PROOF. Given the sentence τ , we construct σ to say, “If I am provable then τ ,” as above. Suppose that T ! ( PrbT τ → τ ). By Formalized Lemma 37C we have T ! ( PrbT σ → PrbT τ ). By our choice of σ , we conclude that T ! σ . So by the (unformalized) Lemma 37C, we have T ! τ . L¨ob’s theorem was originally devised in order to solve the problem given in Exercise 1. But it implies (and in a sense is equivalent to) G¨odel’s second incompleteness theorem. Assume that T is a sufﬁciently strong axiomatizable theory. Applying L¨ob’s theorem and taking τ to be 0 = S0, we have T ! (PrbT (0 = S0) → 0 = S0) ⇒ T ! 0 = S0, that is, T ! Cons T ⇒ T is inconsistent. Thus we obtain a proof of the second incompleteness theorem. But there is an issue not yet examined: What theories are sufﬁciently strong? Are there any at all (apart from the trivial case of the inconsistent theory)? Yes, and here are two. The ﬁrst is called “Peano arithmetic” (PA). Its axioms consist of the A E axioms, plus all the “induction axioms.” These are the universal closures of formulas having the form ϕ(0) ∧ ∀ x(ϕ(x) → ϕ(Sx)) → ∀ x ϕ(x)

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A Mathematical Introduction to Logic for a wff ϕ. The induction axioms — which state the ordinary principle of mathematical induction — enable us to carry out many arguments about the natural numbers (e.g., the commutative law of addition) within Peano arithmetic. But to be sure that formalized reﬂection and formalized modus ponens can be derived with Peano arithmetic, one must carry out the details, which we will not go through here. We know that Peano arithmetic is consistent, because it is true in N. But by the second incompleteness theorem, PA cannot prove its own consistency. We “know” that PA is consistent by means of an argument we carry out either in informal mathematics, or — if we want — in set theory. So set theory has a higher “consistency strength” than PA: It proves the consistency of PA and PA does not. A second sufﬁciently strong theory is axiomatic set theory. Or to be more careful, it is the set of sentences in the language of number theory that are provable in axiomatic set theory. The next subsection deals with this situation. This theory has the advantage that it is quite believable — on an informal level — that formalized reﬂection and formalized modus ponens are derivable. But what are our grounds for thinking that set theory is consistent? We know that PA is consistent because it is true in the “standard model” N of number theory. It is not at all clear that we can meaningfully speak of a “standard model of set theory”!

Applications to Set Theory We know that in the language of number theory, Cn A E is incomplete and nonrecursive, as is any compatible recursively axiomatizable theory in the language. But now suppose we leave arithmetic for a while and look at set theory. Here we have a language (with the parameters ∀ and ∈) and a set of axioms. In all presently accepted cases the set of axioms is recursive. Or more precisely, the set of G¨odel numbers of the axioms is recursive. And so the theory (set theory) obtained is recursively enumerable. We claim that this theory, if consistent, is not recursive and hence not complete. We can already sketch the argument in rough form. We can, in a very real sense, embed the language of number theory in set theory. We can then look at that fragment of set theory which deals with the natural numbers and their arithmetic (the shaded area in Fig. 14). That is a theory compatible with A E . And so it is nonrecursive. Now if set theory were recursive, then its arithmetical part would also be recursive, which it is not. As a bonus, we will come across the second incompleteness theorem for the case of set theory. Henceforth by set theory (ST) we mean that theory (in the language with equality having the two parameters ∀ and ∈) which is the set of consequences of the reader’s favorite set-theoretic axioms. (The standard Zermelo–Fraenkel axioms will do nicely, if the reader has no favorite.

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Sentences in the language of number theory (recursive)

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A recursively enumerable theory which includes AE Set theory (recursively enumerable) (a)

Sentences in the language of number theory (recursive) Image of the above set under an interpretation Set theory (recursively enumerable)

(b)

Figure 14. Set theory and number theory. (a) Flat picture. (b) A more accurate picture.

We ask only that the set of axioms be recursive, and that it be strong enough to yield certain everyday facts about sets.) We need an interpretation π of Cn A E into ST. (The remainder of this section assumes a familiarity with Section 2.7.) But the existence of such a π is a standard result of set theory, although it is not usually stated in these words. We need formulas of the language of ST that adequately express the concept of being a natural number, being the sum of two given numbers, and so forth. To ﬁnd these formulas, we turn to the way in which the arithmetic of natural numbers can be “embedded” in set theory. That is, on the one hand, natural numbers such as 2 or 7 do not appear to be sets. On the other hand, we can, when we choose, select sets to represent numbers. The standard approach is to take 0 to be the set ∅ and n + 1 to be the set n; n. This has the fringe beneﬁt that each number is the set of all smaller numbers (e.g., 3 ∈ 7). Let ω be the collection of all these sets

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A Mathematical Introduction to Logic (these “number-sets”); thus ω is the set representing N. The formula π∀ is the result of eliminating the deﬁned symbol ω from the formula v1 ∈ ω. The formula π0 is similarly obtained from the set-theoretic formula v1 = ∅, and the formula πS is obtained from v2 = v1 ∪ {v1 }. The formula π< is simply v1 ∈ v2 . For π+ we use the translation into the language of ST of For any f,

if f : ω × ω → ω and for all a and b in ω we have f (a, ∅) = a and f (a, b ∪ {b}) = f (a, b) ∪ { f (a, b)}, then f (v1 , v2 ) = v3 .

(The manner of translation is partially indicated in Chapter 0.) The formulas π. and πE are obtained in much the same fashion. The claim that this π is an interpretation of Cn A E into ST makes a number (and the number is 17) of demands on ST. (i) ∃ v1 π∀ must be in ST. It is, since we can prove in set theory that ω is nonempty. (ii) For each of the ﬁve function symbols f in the language of A E , ST must contain a sentence asserting, roughly, that π f deﬁnes a function on the set deﬁned by π∀ . (The exact sentence is set forth in the deﬁnition of interpretation in Section 2.7.) In the case of 0, we have in ST the result that there is a unique empty set and that it belongs to ω. The case for S is simple, since πS deﬁnes a unary operation on the universe of all sets, and ω is closed under this operation. For + we must use the recursion theorem on ω. That is, we can prove in ST (as sketched in Section 1.4) that there is a unique f : ω × ω → ω such that f (a, ∅) = a and f (a, b ∪ {b}) = f (a, b) ∪ { f (a, b)} for a, b in ω. The required property of π+ then follows. Similar arguments apply to · and E. (iii) For each of the 11 sentences σ in A E , the sentence σ π must be in ST. For example, in the case of L3, we have in ST the fact that for any m and n in ω, either m ∈ n, m = n, or n ∈ m. Since these demands are ﬁnite in number, there is a ﬁnite ⊆ ST such that π is also an interpretation of Cn A E into Cn . THEOREM 37D (STRONG UNDECIDABILITY OF SET THEORY) Let T be a theory in the language of set theory such that T ∪ ST (or at least T ∪ ) is consistent. Then T is not recursive. PROOF. Let be the consistent theory Cn(T ∪ ). Let 0 be the corresponding theory π −1 [] in the language of number theory. From Section 2.7 we know that 0 is a consistent theory (since is). Also A E ⊆ 0 , since if σ ∈ A E , then σ π ∈ Cn ⊆ .

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Hence by the strong undecidability of Cn A E (Theorem 35C), 0 is not recursive. Now we must derive the nonrecursiveness of T from that of 0 . We have σ ∈ 0

iff

σπ ∈

and by the lemma below, σ π depends recursively on σ . That is, 0 ≤m . Hence cannot be recursive, lest 0 be. Similarly, we have τ ∈

iff

(ϕ → τ ) ∈ T,

where ϕ is the conjunction of the members of . Since (ϕ → τ ) depends recursively on τ , we have ≤m T so that T cannot be recursive lest be. LEMMA 37E There is a recursive function p such that for any formula α of the language of number theory, p(α) = (α π ). PROOF. In Section 2.7 we gave explicit instructions for constructing α π . The construction in some cases utilized formulas β π for formulas β simpler than α. The methods of Sections 3.3 and 3.4 can be applied to the G¨odel numbers of these formulas to show that p is recursive. But the details are not particularly attractive, and we omit them here. COROLLARY 37F

If set theory is consistent, then it is not complete.

PROOF. Set theory has a recursive set of axioms. If complete, the theory is then recursive (by item 21 of Section 3.4). By the foregoing theorem, this cannot happen if ST is consistent. COROLLARY 37G In the language with equality and a two-place predicate symbol, the set of (G¨odel numbers of) valid sentences is not recursive. PARTIAL PROOF. In the foregoing theorem take T = Cn∅, the set of valid sentences. The theorem then assures us that T is nonrecursive, provided that is consistent. We have not given the ﬁnite set explicitly. But we assure the reader that can be chosen in such a way as to be provably consistent. It should be noted that π is not an interpretation of Th N into ST (unless ST is inconsistent). For π −1 [ST] is a recursively enumerable theory in the language of N, as a consequence of Lemma 37E. Hence it cannot coincide with Th N, and it can include the complete theory Th N only if it is inconsistent.

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Godel’s ¨ Second Incompleteness Theorem for Set Theory We can employ our usual tricks to ﬁnd a sentence a of number theory which indirectly asserts that its own interpretation σ π is not a theorem of set theory. For let D be the ternary relation on N such that a, b, c ∈ D

iff a is the G¨odel number of a formula α of number theory and c is the G¨odel number of a deduction from the axioms of ST of α(Sb 0)π .

The relation D is recursive (by the usual arguments); let δ(v1 , v2 , v3 ) represent D in Cn A E . Let r be the G¨odel number of ∀ v3 ¬ δ(v1 , v1 , v3 ) and let σ be ∀ v3 ¬ δ(Sr 0, Sr 0, v3 ). / ST. We will now prove Observe that σ does indirectly assert that σ π ∈ that the assertion is correct: / ST. LEMMA 37H If ST is consistent, then σ π ∈ PROOF. Suppose to the contrary that σ π is deducible from the axioms of ST; let k be G of such a deduction. Then r, r, k ∈ D. ∴ A E ! δ(Sr 0, Sr 0, Sk 0); ∴ A E ! ∃ v3 δ(Sr 0, Sr 0, v3 ); i.e., A E ! ¬ σ. Applying our interpretation π , we conclude that ¬ σ π is in ST, whence ST is inconsistent. Thus / ST. ST is consistent ⇒ σ π ∈

Now the above proof, like all those in this book, is carried out in informal mathematics. But all of our work in the book could have been carried out within ST. Indeed it is common knowledge that essentially all work in mathematics can be carried out in ST. Imagine actually doing so. Then instead of a proof of an English sentence, “ST is consistent ⇒ σπ ∈ / ST,” we have a deduction from the axioms of ST of a certain sentence in the formal language of set theory: (Cons(ST) → ). Here Cons(ST) is the result of translating (in a nice way) “ST is consistent” into the language of set theory. Similarly, is the result of translating “σ π ∈ / ST.” But we already have a sentence in the language

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of set theory asserting that σ π ∈ / ST. It is σ π . This strongly suggests that is (or is provably equivalent in ST to) σ π , from which we get (Cons(ST) → σ π ) as a theorem of ST. Now this can actually be carried out in such a way as to have be σ π . We have given above an argument, which we hope will convince the reader that this is at least probable. And from it we now have the result: GODEL ¨ ’S SECOND INCOMPLETENESS THEOREM FOR SET THEORY The sentence Cons(ST) is not a theorem of ST, unless ST is inconsistent. PROOF.

By the above (plausibility) argument (Cons(ST) → σ π )

is a theorem of ST. So if Cons(ST) is also a theorem of ST, then σ π is, too. But by Lemma 37H, if σ π ∈ ST, then ST is inconsistent. Of course if ST is inconsistent, then every sentence is a theorem, including Cons(ST). Because of this, a proof of Cons(ST) within ST would not convince people that ST was consistent. (And by G¨odel’s second theorem, it would convince them of the opposite.) But prior to G¨odel’s work it was possible to hope that Cons(ST) might be provable from assumptions weaker than the axioms of set theory, ideally assumptions already known to be consistent. But we now see that Cons(ST) is not in any subtheory of ST, unless of course ST is inconsistent. We are left with the conclusion that any recursively axiomatizable theory of sets (provided it meets the desirable conditions of being consistent and strong enough to prove everyday facts) is an incomplete theory. This raises a challenge: to ﬁnd additional axioms to add to the theory. On the one hand, we want the additional axioms to strengthen the theory in useful ways. On the other hand, we want the additional axioms to reﬂect accurately our informal ideas about what sets really are and how they really behave.

Exercises 1. Let σ be a sentence such that A E ! (σ ↔ Prb A E σ ). (Thus σ says “I am provable,” in contrast to the sentence “I am unprovable” that has been found to have such interesting properties.) Does A E ! σ ?

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A Mathematical Introduction to Logic 2. Let T be a theory in a recursively numbered language, and assume that there is an interpretation of Cn A E into T . Show that T is strongly undecidable; i.e., whenever T is a theory in the language for which T ∪ T is consistent, then T is not recursive.

SECTION 3.8 Representing Exponentiation1 In Sections 3.1 and 3.2 we studied the theory of certain reducts of N and found them to be decidable. Then in Section 3.3 we added both multiplication and exponentiation. The resulting theory was found (in Section 3.5) to be undecidable. Actually it would have been enough to add only multiplication (and forego exponentiation); we would still have undecidability. Let N M be the reduct of N obtained by dropping exponentiation: N M = (N; 0, S, <, +, ·). Thus the symbol E does not appear in the language of N M . Let A M be the set obtained from A E by dropping E1 and E2. The purpose of this section is to show that all the theorems of Sections 3.3–3.5 continue to hold when “A E ” and “N” are replaced by “ A M ” and “N M .” The key fact needed to establish this claim is that exponentiation is representable in Cn A M . That is, there is a formula ε in the language of N M such that for any a and b, A M ! ∀ z[ε(Sa 0, Sb 0, z) ↔ z = S(a ) 0]. b

Thus ε(x, y, z) can be used to simulate the formula xEy = z without actual use of the symbol E. If we look to see what relations and functions are representable in Cn A M , we ﬁnd at ﬁrst that everything (except for exponentiation itself) that was shown to be representable in Cn A E is (by the same proof) representable in Cn A M . Until, that is, we reach item 7 in the catalog listing of Section 3.3. To go further, we must show that exponentiation itself is representable in Cn A M . We know that exponentiation can be characterized by the recursion equations a 0 = 1, a b+1 = a b · a.

1

This section may be omitted without loss of continuity.

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From what we know about primitive recursion (catalog item 13 in Section 3.3 plus Exercise 8 there), we might think of deﬁning E ∗ (a, b) = the least s such that [(s)0 = 1 and for all i < b, (s)i+1 = (s)i · a]. For then a b = (E ∗ (a, b))b . This fails to yield a proof of representability, because we do not yet know that the decomposition function (a)b is representable in Cn A M . But we do not really need that particular decomposition function (which corresponded to a particular way of encoding sequences). All we need is some function δ that acts like a decomposition function; the properties we need are summarized in the following lemma. LEMMA 38A There is a function δ representable in Cn A M such that for every n, a0 , . . . , an , there is an s for which δ(s, i) = ai for all i ≤ n. Once the lemma has been established, we can deﬁne E ∗∗ (a, b) = the least s such that [δ(s, 0) = 1 and for all i < b, δ(s, i + 1) = δ(s, i) · a]. The lemma assures us that such an s exists. E ∗∗ is then representable in Cn A M , as is exponentiation, since a b = δ(E ∗∗ (a, b), b). A function δ that establishes the lemma will be provided by some facts of number theory.

A Pairing Function As a ﬁrst step toward proving the foregoing lemma, we will construct functions for encoding and decoding pairs of numbers. It is well known that there exist functions mapping N × N one-to-one onto N. In particular, the function J does this, where in the diagram shown, J (a, b) has been written at the point with coordinates a, b .

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A Mathematical Introduction to Logic For example, J (2, 1) = 8 and J (0, 2) = 3. To obtain an equation for J (a, b), we note that along the line x + y = n there are n + 1 points (with coordinates in N). Thus J (a, b) = the number of points in the plane to which J assigns smaller values = [the number of points on lines x + y = n for n = 0, 1, . . . , (a + b − 1)] + [the number of points on the line x + y = a + b for which x < a] = [1 + 2 + · · · + (a + b)] + a = 12 (a + b)(a + b + 1) + a = 12 [(a + b)2 + 3a + b]. Let K and L be the corresponding projection functions onto the axes, i.e., the unique functions such that K (J (a, b)) = a,

L(J (a, b)) = b.

For example, K (7) = 1, the x-coordinate of the point 1, 2 in the plane to which J assigned the number 7. Similarly, L(7) = 2, the y-coordinate of that point. We claim that J, K , and L are representable in Cn A M . The function H (a) = the least b such that a ≤ 2b has the property that H (a) = 12 a for even a. Then we can write J (a, b) = H ((a + b) · (a + b + 1)) + a, K ( p) = the least a such that [for some b ≤ p, J (a, b) = p], L( p) = the least b such that [for some a ≤ p, J (a, b) = p]. From the form of the four preceding equations we conclude that H , J , K , and L are representable in Cn A M .

The Godel ¨ β -function Let β be the function deﬁned as follows: β(c, d, i) = the remainder in c ÷ [1 + (i + 1) · d] = the least r such that for some q ≤ c, c = q · [1 + (i + 1) · d] + r. This unlikely-looking function produces a satisfactory decomposition function for Lemma 38A. Let δ(s, i) = β(K (s), L(s), i). It is clear that δ is representable in Cn A M . What is not so obvious is that it meets the conditions of Lemma 38A. We want to show: For any n and any a0 , . . . , an , there are numbers (∗) c and d such that β(c, d, i) = ai for all i ≤ n. For then it follows that δ(J (c, d), i) = β(c, d, i) = ai for i ≤ n.

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Now (∗) is a statement of number theory, not logic. The proof of (∗) is based on the Chinese remainder theorem. Numbers d0 , . . . , dn are said to be relatively prime in pairs iff no prime divides both di and d j for i =

j. CHINESE REMAINDER THEOREM Let d0 , . . . , dn , be relatively prime in pairs; let a0 , . . . , an be natural numbers with each ai < di . Then we can ﬁnd a number c such that for all i ≤ n, ai = the remainder in c ÷ di . PROOF. Let p = i≤n di , and for any c let F(c) be the (n + 1)-tuple of remainders when c is divided by d0 , . . . , dn . Notice that there are p possible values for this (n + 1)-tuple. We claim that F is one-to-one on {k | 0 ≤ k < p}. For suppose that F(c1 ) = F(c2 ). Then each di divides |c1 − c2 |. Since the di ’s are relatively prime, p must divide |c1 − c2 |. For c1 , c2 less than p, this implies that c1 = c2 . Hence the restriction of F to {k | 0 ≤ k < p} takes on all p possible values. In particular, it assumes (at some point c) the value a0 , . . . , an . And that is the c we want. LEMMA 38B

For any s ≥ 0, the s + 1 numbers

1 + 1 · s!, 1 + 2 · s!, . . . , 1 + (s + 1) · s! are relatively prime in pairs. PROOF. All these numbers have the property that any prime factor q cannot divide s!, whence q > s. If the prime q divides both 1 + j · s! and 1 + k · s!, then it divides their difference, | j − k| · s!. Since q does not divide s!, it divides | j − k|. But | j − k| ≤ s < q. This is possible only if | j − k| = 0. PROOF OF (∗). Assume we are given a0 , . . . , an ; we need numbers c and d such that the remainder when c is divided by 1 + (i + 1) · d is ai , for i ≤ n. Let s be the largest of {n, a0 , . . . , an } and let d = s!. Then by Lemma 38B, the numbers 1 + (i + 1) · d are relatively prime in pairs for i ≤ n. So by the Chinese remainder theorem there is a c such that the remainder in c ÷ [1 + (i + 1) · d] is ai for i ≤ n. This completes the proof of Lemma 38A. And by the argument that followed that lemma, we can conclude: THEOREM 38C

Exponentiation is representable in Cn A M .

Armed with this theorem, we can now return to catalog item 7 of Section 3.3. The proof given there now establishes that the function in question (whose value at n is pn ) is representable in Cn A M . For it was

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Structure

Theory

(N)

Decidable. Not ﬁnitely Any inﬁnite set. axiomatizable. Admits elimination of quantiﬁers.

∅ and N.

As above.

Any inﬁnite set with distinguished element.

∅, {0}, N − {0}, N.

(N; 0)

Models of the theory

Deﬁnable sets

Comments

{0} is not deﬁnable.

S is not deﬁnable.

(N; 0, S)

As above.

Standard part plus any number of Z-chains.

Finite and coﬁnite sets. < is not deﬁnable.

{0} is deﬁnable in (N; S).

(N; 0, S, <)

Decidable. Finitely axiomatizable. Admits elimination of quantiﬁers.

As above, with any ordering of the Z-chains.

Finite and coﬁnite sets. + is not deﬁnable.

{0} and S are deﬁnable in (N; <).

(N; 0, S, <, +)

Decidable (Presburger).

The Z-chains are densely ordered without endpoints. Also there is a suitable addition operation.

Eventually periodic sets. · is not deﬁnable.

{0}, S, and < are deﬁnable in (N; +).

Not arithmetical.

As above, but with a suitable multiplication operation.

All arithmetical relations are deﬁnable.

The arithmetical relations are deﬁnable in (N; S, ·), (N; +, ·), and (N; <, D), where D(x, y) = (x) y .

(N; 0, S, <, +, ·)

∴ not recursively axiomatizable.

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formed by allowable methods from relations and functions (including exponentiation) known to be representable in Cn A M . The same phenomenon persists throughout Sections 3.3 and 3.4. The representability proofs given there now establish representability in Cn A M . Thus any recursive relation is representable in Cn A M , and if the relation happens to be a function, then it is functionally representable. The proofs given in Section 3.5 then apply to N M and A M as well as to N and A E . In particular, we have the strong undecidability of Cn A M : Any theory T in the language of N M for which T ∪ A M is consistent cannot be recursive. Notice that any relation deﬁnable in N (i.e., any arithmetical relation) is also deﬁnable in N M . For exponentiation, being representable in a subtheory of Th N M , is a fortiori deﬁnable in N M . By the new version of Tarski’s theorem, ThN M is not deﬁnable in N M , and consequently ThN M cannot be arithmetical. In the terminology of Section 2.7, we can say that there is a faithful interpretation of Th N into Th N M . It equals the identity interpretation on all parameters except E, and to E it assigns a formula deﬁning exponentiation in N M . In Table X we summarize some of the results of Chapter 3 on number theory and its reducts.

Exercises 1. Let D(a, b) = (a)b . Show that any arithmetical relation is deﬁnable in the structure (N; <, D). Remark: One may well ask why Th N A , arithmetic with addition, is decidable (as shown in Section 3.2), while Th N M , the theory of arithmetic with addition and multiplication, is undecidable. One answer is that, as this section shows, multiplication lets us do a certain amount of sequence coding and decoding. The point of this exercise is to show that once we have the decoding function D and ordering, we have the full complexity of arithmetic with addition, multiplication, and exponentiation. 2. Show that the addition relation { a, b, c | a + b = c} is deﬁnable in the structure (N; S, ·). Suggestion: Under what conditions does the equation S(ac) · S(bc) = S(c · c · S(ab)) hold? 3. (a) Show that Th(Z; +, ·) is strongly undecidable. (See Exercise 2 of Section 3.7.) (b) (This part assumes a background in algebra.) Show that the theory of rings is undecidable and that the theory of commutative rings is undecidable.

Chapter

F O U R

Second-Order Logic SECTION 4.1 Second-Order Languages We can obtain — at a cost — richer, more expressive languages than the ﬁrst-order languages considered thus far, by allowing quantiﬁcation of predicate or function symbols. For example, ∃ x(P x → ∀ x P x) is a valid formula having ∀ and P as its parameters. Because it is true no matter how P is interpreted, ∀ P ∃ x(P x → ∀ x P x) deserves to be called valid. (Now ∀ is the only parameter, since P is treated as a predicate variable.) Suppose, then, that we have in addition to the symbols introduced at the beginning of Section 2.1, the further logical symbols: 4. Predicate variables: For each positive integer n we have the n-place predicate variables Xn1 , Xn2 , . . . . 5. Function variables: For each positive integer n, we have the n-place function variables Fn1 , Fn2 , . . . . The usual variables v1 , v2 , . . . will now be called individual variables, to avoid confusion. The terms are as before deﬁned as the expressions that can be built up from the constant symbols and the 282

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individual variables by applying the function symbols (both the function parameters and the function variables). Atomic formulas are again expressions Pt1 · · · tn , where t1 , . . . , tn are terms and P is an n-place predicate symbol (parameter or variable). The deﬁnition of wff is augmented by new formula-building operations: If ϕ is a wff, then so also are ∀ Xin ϕ and ∀ Fin ϕ. The concept of a variable occurring free in ϕ is deﬁned just as before. A sentence is a wff σ in which no variable (individual, predicate, or function) occurs free. It should be remarked that the roles played by predicate parameters and free predicate variables are essentially the same. There is the same close relationship between constant symbols and free individual variables, and between function parameters and free function variables. By a structure we continue to mean a function on the set of parameters meeting the conditions set forth in Section 2.2. We must extend the deﬁnition of satisfaction in the natural way. Let V now be the set of all variables, individual, predicate, or function. Let s be a function on V that assigns to each variable the suitable type of object. Thus s(v1 ) is a member of the universe, s(Xn ) is an n-ary relation on the universe, and s(Fn ) is an n-ary operation. For a term t, s(t) is deﬁned in the natural way. In particular, if F is a function variable, then s(Ft1 · · · tn ) is the result of applying the function s(F) to s(t1 ), . . . , s(tn ) . Satisfaction of atomic formulas is also deﬁned essentially as before. For a predicate variable X, |=A Xt1 · · · tn [s]

iff

s(t1 ), . . . , s(tn ) ∈ s(X).

The only new features in the deﬁnition of satisfaction arise from our new quantiﬁers. 5. |=A ∀ Xin ϕ[s] iff for every n-ary relation R on |A|, we have |=A ϕ[s(Xin | R)]. 6. |=A ∀ Fin ϕ[s] iff for every function f : |A|n → |A|, we have |=A ϕ[s(Fin | f )]. Again it is easy to see that only the values of s at variables occurring free in the formula are signiﬁcant. For a sentence σ , we may unambiguously speak of its being true or false in A. Logical (semantical) implication is deﬁned exactly as before. EXAMPLE 1. A well-ordering is an ordering relation such that any nonempty set has a least (with respect to the ordering) element. This last condition can be translated into the second-order sentence ∀ X( ∃ y Xy → ∃ y(Xy ∧ ∀ z(Xz → y ≤ z))). Here, as elsewhere, we omit the subscripts on X and F when they are immaterial, and we omit the superscripts if they are clear from the context.

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A Mathematical Introduction to Logic EXAMPLE 2. One of Peano’s postulates (the induction postulate) states that any set of natural numbers that contains 0 and is closed under the successor function is, in fact, the set of all natural numbers. This can be translated into the second-order language for number theory as ∀ X(X0 ∧ ∀ y(Xy → XSy) → ∀ y Xy). Any model of S1, S2, and the above Peano induction postulate is isomorphic to (N; 0, S); see Exercise 1. Thus this set of sentences is categorical; i.e., all its models are isomorphic. EXAMPLE 3. For any formula ϕ in which the predicate variable Xn does not occur free, the formula ∃ Xn ∀ v1 · · · ∀ vn [Xn v1 · · · vn ↔ ϕ] is valid. (Here other variables may occur free in ϕ in addition to v1 , . . . , vn .) It says that there exists a relation consisting of exactly the n-tuples satisfying ϕ. Formulas of this form are called relation comprehension formulas. There are also the analogous function comprehension formulas. If ψ is a formula in which the variable Fn does not occur free, then ∀ v1 · · · ∀ vn ∃!vn+1 ψ → ∃ Fn ∀ v1 · · · ∀ vn+1 (Fn v1 · · · vn = vn+1 ↔ ψ) is valid. (Here “∃!vn+1 ψ” is an abbreviation for a formula obtained from Exercise 21 of Section 2.2.) EXAMPLE 4. In the ordered ﬁeld of real numbers, any bounded nonempty set has a least upper bound. We can translate this by the second-order sentence ∀ X[∃ y ∀ z(Xz → z ≤ y) ∧ ∃ z Xz → ∃ y ∀ y ( ∀ z(Xz → z ≤ y ) ↔ y ≤ y )]. It is known that any ordered ﬁeld that satisﬁes this second-order sentence is isomorphic to the ordered ﬁeld of reals. EXAMPLE 5. For each n ≥ 2, we have a ﬁrst-order sentence λn which translates, “There are at least n things.” For example, λ3 is ∃ x ∃ y ∃ z(x = y∧x= z ∧ y= z). The set {λ2 , λ3 , . . .} has for its class of models the EC class consisting of the inﬁnite structures. There is a single secondorder sentence that is equivalent. A set is inﬁnite iff there is an ordering on it having no last element. Or more simply, a set is

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inﬁnite iff there is a transitive irreﬂexive relation R on the set whose domain is the entire set. This condition can be translated into a second-order sentence λ∞ : ∃ X[∀ u ∀ v ∀ w(Xuv → Xvw → Xuw) ∧ ∀ u ¬ Xuu ∧ ∀ u ∃ v Xuv]. Another sentence (using a function variable) that deﬁnes the class of inﬁnite structures is ∃ F[∀ x ∀ y(Fx = Fy → x = y) ∧ ∃ z ∀ x Fx = z], which says there is a one-to-one function that is not onto. The preceding example shows that the compactness theorem fails for second-order logic: THEOREM 41A There is an unsatisﬁable set of second-order sentences every ﬁnite subset of which is satisﬁable. PROOF.

The set is, in the notation of the above example, {¬ λ∞ , λ2 , λ3 , . . .}.

The L¨owenheim–Skolem theorem also fails for second-order logic. By the language of equality we mean the language (with =) having no parameters other than ∀. A structure for this language can be viewed as being simply a nonempty set. In particular, a structure is determined to within isomorphism by its cardinality. A sentence in this language is therefore determined to within logical equivalence by the set of cardinalities of its models (called its spectrum). THEOREM 41B There is a sentence in the second-order language of equality that is true in a set iff its cardinality is 2ℵ0 . PROOF, USING CONCEPTS FROM ALGEBRA AND ANALYSIS. Consider ﬁrst the conjunction of the (ﬁrst-order) axioms for an ordered ﬁeld, further conjoined with the second-order sentence expressing the least-upper-bound property (see Example 4 of this section). This is a sentence whose models are exactly the isomorphs of the real ordered ﬁeld (i.e., the structures isomorphic to the ordered ﬁeld of real numbers). We now convert the parameters 0, 1, +, ·, < to variables (individual, function, or predicate as appropriate) which we existentially quantify. The resulting sentence has the desired properties. There are other cardinal numbers with second-order characterizations of this sort; cf. Exercise 2.

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A Mathematical Introduction to Logic THEOREM 41C The set of G¨odel numbers of valid second-order sentences is not deﬁnable in N by any second-order formula. Here we assume that G¨odel numbers have been assigned to secondorder expressions in a manner like that used before. Although our proof applies to the second-order language of number theory, the theorem is true for any recursively numbered language having at least a two-place predicate symbol. PROOF. Let T 2 be the second-order theory of N, i.e., the set of second-order sentences true in N. The same argument used to prove Tarski’s theorem shows that T 2 is not deﬁnable in N by any second-order formula. Now let α be the conjunction of the members of A E with the second-order Peano induction postulate (Example 2). Any model of α is isomorphic to N; cf. Exercise 1. Consequently, for any sentence σ , σ ∈ T2

iff

(α → σ ) is valid.

Consequently, the set of (G¨odel numbers of) validities cannot be deﬁnable lest T 2 be. A fortiori, the set of G¨odel numbers of second-order validities is not arithmetical and not recursively enumerable. That is, the enumerability theorem fails for second-order logic. (In the other direction, one can show that this set is not deﬁnable in number theory of order three, or even of order ω. But these are topics we will not enter into here.) It is interesting to compare the effect of a second-order universal sentence, such as the Peano induction postulate ∀ X(X0 ∧ ∀ y(Xy → XSy) → ∀ y Xy) and the corresponding ﬁrst-order “schema,” i.e., the set of all sentences ϕ(0) ∧ ∀ y(ϕ(y) → ϕ(Sy)) → ∀ y ϕ(y), where ϕ is a ﬁrst-order formula having just v1 free. If A is a model of the Peano induction postulate, then any subset of |A| containing 0A and closed under SA is in fact all of |A|. On the other hand, if A is a model of the corresponding axiom schema, we can say only that every deﬁnable subset of |A| containing 0A and closed under SA is all of |A|. There may well be undeﬁnable subsets for which this fails. (For example, take any model A of Th(N; 0, S) having Z-chains. Then A satisﬁes the above ﬁrst-order schema, but it does not satisfy the second-order induction postulate. The set of standard points is simply not deﬁnable in N.)

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Exercises 1. Show that any structure for the language with parameters ∀, 0, and S that satisﬁes the sentences ∀ x Sx = 0

(S1)

∀ X ∀ y(Sx = Sy → x = y)

(S2)

and the Peano induction postulate ∀ X(X0 ∧ ∀ y(Xy → XSy) → ∀ y Xy) is isomorphic to N S = (N; 0, S). 2. (a) Give a sentence in the second-order language of equality that is true in a set iff its cardinality is ℵ0 . (b) Do the same for ℵ1 . 3. Let ϕ be a formula in which only the n-place predicate variable X occurs free. Say that an n-ary relation R on |A| is implicitly deﬁned in A by ϕ iff A satisﬁes ϕ with an assignment of R to X but does not satisfy ϕ with an assignment of any other relation to X. Show that Th N, the set of G¨odel numbers of ﬁrst-order sentences true in N, is implicitly deﬁnable in N by a formula without quantiﬁed predicate or function variables. Suggestion: The idea is to write down conditions that the set of true sentences must meet. 4. Consider a language (with equality) having the one-place predicate symbols I and S and the two-place predicate symbol E. Find a second-order sentence σ such that (i) if A is a set for which A ∩ P A = ∅ and if |A| = A ∪ P A, I A = A, S A = P A, E A = { a, b | a ∈ b ⊆ A}, then A is a model of σ ; and (ii) every model of σ is isomorphic to one of the sort described in (i). Remark: Roughly speaking, σ translates “S = P I .”

SECTION 4.2 Skolem Functions We want to show how, given any ﬁrst-order formula, one can ﬁnd a logically equivalent prenex second-order formula of a very special form:

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A Mathematical Introduction to Logic This is a prenex formula wherein all universal quantiﬁers are individual ones that follow a string of existential individual and function quantiﬁers. In the simplest example, observe that ∀ x ∃ y ϕ(x, y) |==| ∃ F ∀ x ϕ(x, Fx). In the “=|” direction this is easy to see. For the “|=” direction, consider a structure A and an assignment function s satisfying ∀ x ∃ y ϕ(x, y). We know that for any a ∈ |A| there is at least one b ∈ |A| such that |=A ϕ(x, y)[s(x | a)(y | b)]. We obtain a function f on |A| by choosing one such b for each a and taking f (a) = b. (The axiom of choice is used here.) Then |=A ∀ x ϕ(x, Fx)[s(F | f )]. This function f is called a Skolem function for the formula ∀ x ∃ y ϕ in the structure A. The same argument applies more generally. As a second example, suppose that we begin with the formula ∃ y1 ∀ x1 ∃ y2 ∀ x2 ∀ x3 ∃ y3 ψ(y1 , y2 , y3 ). (We have listed only y1 , y2 , and y3 , but possibly other variables occur free in ψ as well.) Here we already have the existential quantiﬁer ∃ y1 at the left. What remains is ∀ x1 ∃ y2 ∀ x2 ∀ x3 ∃ y3 ψ(y1 , y2 , y3 ). This is a special case of the ﬁrst example (with ϕ(x1 , y2 ) = ∀ x2 ∀ x3 ∃ y3 ψ(y1 , y2 , y3 )). It is logically equivalent, as before, to ∃ F2 ∀ x1 ∀ x2 ∀ x3 ∃ y3 ψ(y1 , F2 x1 , y3 ). Now we have the existential quantiﬁers ∃ y1 ∃ F2 at the left; what remains is ∀ x1 ∀ x2 ∀ x3 ∃ y3 ψ(y1 , F2 x1 , y3 ). By the same reasoning as before, this is logically equivalent to ∃ F3 ∀ x1 ∀ x2 ∀ x3 ψ(y1 , F2 x1 , F3 x1 x2 x3 ), where F3 is a three-place function variable. Thus the original formula is equivalent to ∃ y1 ∃ F2 ∃ F3 ∀ x1 ∀ x2 ∀ x3 ψ(y1 , F2 x1 , F3 x1 x2 x3 ). For quantiﬁer-free ψ, this is in the form we desire. SKOLEM NORMAL FORM THEOREM For any ﬁrst-order formula, we can ﬁnd a logically equivalent second-order formula consisting of:

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(a) First a string (possibly empty) of existential individual and function quantiﬁers, followed by (b) A string (possibly empty) of universal individual quantiﬁers, followed by (c) A quantiﬁer-free formula. A formal proof could be given using induction, but the preceding example illustrates the general method. Recall that a universal (∀1 ) formula is a ﬁrst-order prenex formula all of whose quantiﬁers are universal: ∀ x1 ∀ x2 · · · ∀ xk α, where α is quantiﬁer-free. Similarly, an existential (∃1 ) formula is a ﬁrst-order prenex formula all of whose quantiﬁers are existential. COROLLARY 42A For any ﬁrst-order ϕ, we can ﬁnd a universal formula θ in an expanded language containing function symbols, such that ϕ is satisﬁable iff θ is satisﬁable. By applying this corollary to ¬ ϕ, we obtain an existential formula (with function symbols) that is valid iff ϕ is valid. PROOF. Again we will only illustrate the situation by an example. Say that ϕ is ∃ y1 ∀ x1 ∃ y2 ∀ x2 ∀ x3 ∃ y3 ψ(y1 , y2 , y3 ). First we replace ϕ by the logically equivalent formula in Skolem form: ∃ y1 ∃ F2 ∃ F3 ∀ x1 ∀ x2 ∀ x3 ψ(y1 , F2 x1 , F3 x1 x2 x3 ). Then for θ we take ∀ x1 ∀ x2 ∀ x3 ψ(c, f x1 , gx1 x2 x3 ), where c, f , and g are new function symbols having zero, one, and three places, respectively. In general θ is not logically equivalent to ϕ. But we do have θ |= ϕ (in the expanded language). And any model A of ϕ can be expanded (by deﬁning cA , f A , and g A correctly) to be a model of θ. Thus ϕ and θ are “equally satisﬁable.” This result reduces the general problem of testing ﬁrst-order formulas for satisﬁability to the special case of universal formulas (with function symbols). And by the same token, it reduces the problem of testing for validity to the ∃1 case. From these reductions we can derive an undecidability result for ﬁrst-order logic: COROLLARY 42B Consider a recursively numbered language having a two-place predicate symbol and inﬁnitely many k-place function symbols for each k ≥ 0.

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A Mathematical Introduction to Logic (a) The set of G¨odel numbers of satisﬁable universal (ﬁrst-order) sentences is not recursive. (b) The set of G¨odel numbers of valid existential (ﬁrst-order) sentences is not recursive. PROOF. (b) Given any sentence σ we can, by applying Corollary 42A to ¬ σ , effectively ﬁnd an existential sentence that is valid iff σ is valid. Hence a decision procedure for the existential validities would yield a decision procedure for arbitrary validities, in contradiction to Church’s theorem. We can use predicate variables instead of function variables in these results, but at a price. Suppose we begin with a ﬁrst-order formula. It is equivalent to a formula ψ in Skolem normal form; suppose for simplicity that ψ = ∃ F ϕ, where ϕ has only individual quantiﬁers and F is a one-place function variable. We can choose ϕ in such a way that F occurs only in equations of the form u = Ft (for terms t and u not containing F). This can be done by replacing, for example, an atomic formula α(Ft) by either ∀ x(x = Ft → α(x)) or ∃ x(x = Ft ∧ α(x)). Next observe that a formula ∃ F u = Ft

,

wherein F occurs only in the form shown, is equivalent to ∃ X( ∀ y ∃!z Xyz ∧

Xtu

).

If one pursues this question (as we will not do here) one ﬁnds that any ﬁrst-order formula is logically equivalent to a second-order formula consisting of (a) (b) (c) (d)

A string of existential predicate quantiﬁers, followed by A string of universal individual quantiﬁers, followed by A string of existential individual quantiﬁers, followed by A quantiﬁer-free formula.

There are corresponding versions — see Exercise 4 — of Corollaries 42A and 42B. The analogue of Corollary 42A reduces the problem of testing ﬁrst-order formulas for satisﬁability to the special case of ∀2 formulas (with predicate symbols). The problem of testing for validity is reduced to the ∃2 case. The analogue of Corollary 42B can be compared with Exercise 10 in Section 2.6, where it is shown that the set of ∀2 validities without function symbols is decidable.

Herbrand Expansions We have seen (in Corollary 42A) how, given a formula of ﬁrst-order logic, to ﬁnd an “equally satisﬁable” universal formula. And thus

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(Corollary 42B) the question of satisﬁability in ﬁrst-order logic is reducible to the question of satisﬁability of universal formulas. Now we go one step further: satisﬁability of these universal formulas is reducible — but in a weaker sense — to satisﬁability in sentential logic. EXAMPLE. We know that ∀ x ∃ y P x y |= ∃ y ∀ x P x y. But pretend we did not know this, and that we are interested in determining whether or not logic implication holds here. That is equivalent to determining whether or not the hypothesis ∀ x ∃ y P x y together with the negation of the conclusion ¬ ∃ y ∀ x P x y is unsatisﬁable. By the Skolem normal form theorem, we can replace these sentences by certain logically equivalent sentences; we wish to determine whether or not ∃ F ∀ x P xFx together with ∃ G ∀ y ¬ PGyy is unsatisﬁable. And as in Corollary 42A, we replace these sentences by equally satisﬁable universal sentences; we wish to determine whether or not the set {∀ x P x f x, ∀ y Pgy y} is unsatisﬁable (where f and g are new function symbols). But this set of universal sentences is satisﬁable, and moreover, the set can be made to generate its own model. Here is how. For the universe of our model we will use the Herbrand universe H , which is the set of all terms (in the language with f and g). Thus H contains, for each variable u, the terms u, f u, gu, f f u, f gu, . . . . Let be the set of all instances of the universal sentences, that is, the formulas obtained by dropping all the universal quantiﬁers and plugging in (for the universally quantiﬁed variables) arbitrary terms from the Herbrand universe. Thus contains, for each variable u, the quantiﬁer-free formulas Pu f u, Pgu f gu, . . . , ¬ Pgu u, ¬ Pg f u f u, . . . . Now we consider from the point of view of sentential logic. The sentence symbols are the atomic formulas, e.g., Pg f u f u. And in this example is satisﬁable in sentential logic. That is, there is a truth assignment v on the set of sentence symbols so that v(α) = T for every α in . Here is one such v: T if t1 is shorter than t2 v(Pt1 t2 ) = F if t1 is at least as long as t2 Finally, we use this truth assignment v (in sentential logic) to make a structure H (in ﬁrst-order logic) that will be a model of the universal sentences. The universe is the Herbrand universe:

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A Mathematical Introduction to Logic |H| = H . (There are echos of the completeness proof of Section 2.5 here.) The function symbols are interpreted autonymously — as naming themselves: f H (t) is f t and g H (t) is gt. Where v comes in is to interpret the predicate symbol P: t1 , t2 ∈ P H ⇐⇒ v(Pt1 t2 ) = T This structure works. First |=H ∀ x P x f x because for every term t in the Herbrand universe, t, f t ∈ P H . Secondly |=H ∀ y ¬ Pgy y because for every term t in the Herbrand universe, gt, t ∈ / P H. We conclude that the hypothesis ∀ x ∃ y P x y together with the negated conclusion ¬ ∃ y ∀ x P x y is indeed satisﬁable, and hence ∀ x ∃ y P x y |= ∃ y ∀ x P x y. To what extent can we generalize from this example? Assume, for simplicity, that the language does not contain equality. (Exercise 7 indicates the changes needed to accommodate equality.) Suppose we want to determine whether or not |= ϕ for a set ; ϕ of formulas in ﬁrstorder logic. This is equivalent to determining whether or not the set ; ¬ ϕ is unsatisﬁable. We can replace each of the formulas here by a logically equivalent formula in Skolem normal form. And as in Corollary 42A, we get an equally satisﬁable set of universal formulas. (In applying Skolem normal form, we use different Skolem function symbols for each formula, so that there is no clash between formulas.) This brings us to the situation: |= ϕ ⇐⇒ is unsatisﬁable and is a set of universal formulas. Let H be the Herbrand universe, that is, the set of all terms in the language of . Let be the set of all instances of the universal formulas in (i.e., formulas obtained by dropping all the universal quantiﬁers and plugging in for the universally quantiﬁed variables arbitrary terms from the Herbrand universe). Then consists of quantiﬁer-free formulas only. We consider from the viewpoint of sentential logic, where the sentence symbols are the atomic formulas. Case I: is unsatisﬁable in sentential logic. In this case, we can conclude that is unsatisﬁable and |= ϕ in ﬁrst-order logic. This is because a universal formula logically implies all of its instances. Therefore |= δ for every δ in (in ﬁrst-order logic). Any model of must be a model of . But from a model A of we can extract a truth assignment v that satisﬁes in sentential logic. (Remember Exercise 3 in Section 2.4. Note the interesting interplay between ﬁrst-order logic and sentential logic.)

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Case II: is satisﬁable in sentential logic, say by the truth assignment v. Then we will use v to make a structure H in which is satisﬁable and |= ϕ because H will give a counterexample. As in the example, the universe |H| is the Herbrand universe H , the set of all terms in the language of . And again the function symbols are interpreted autonymously: f H (t1 , . . . , tn ) = f t1 · · · tn . To interpret a predicate symbol P we use the truth assignment v: t1 , . . . , tn ∈ P H ⇐⇒ v(Pt1 · · · tn ) = T Then we claim that every formula in is satisﬁed in H by the identity function s(x) = x on the variables. First, note that s(t) = t for any term t in H ; we had the same situation in step 4 of the completeness proof in Section 2.5. Secondly, note that for an atomic formula Pt1 · · · tn , |=H Pt1 · · · tn [s] ⇐⇒ t1 , . . . , tn ∈ P H ⇐⇒ v(Pt1 · · · tn ) = T. By Exercise 3 of Section 2.4 again, any formula δ in is satisﬁed in H with s (because v(δ) = T ). Consider any formula in . It is a universal formula; to simplify the notation, say it is ∀ v1 ∀ v2 θ(v1 , v2 , v3 ), where θ is quantiﬁer-free. We need to check for any terms t1 and t2 in H that |=H θ [[t1 , t2 , v3 ]]. This is equivalent (by the substitution lemma) to saying that the formula θ (t1 , t2 , v3 ) is satisﬁed in H by s. But this formula is an instance of ∀ v1 ∀ v2 θ (v1 , v2 , v3 ), and so θ(t1 , t2 , v3 ) is in . As noted above, our construction was arranged so that every formula in is satisﬁed in H with s. This is what we needed. We can summarize the result as follows. For simplicity, the result is stated only for sentences. HERBRAND’S THEOREM Consider a set ; ϕ of sentences in a ﬁrstorder language without equality. Let be as above. Then either (case I) is unsatisﬁable in sentential logic and |= ϕ, or (case II) is satisﬁable in sentential logic and the structure H constructed above is a model of in which ϕ is false. (Herbrand’s work was in his 1930 dissertation, completed not long before he was killed in a mountain-climbing accident. His statement of the theorem was quite different from this one, but the ideas follow his work, and 1928 work of Thoralf Skolem.) In case I, by the compactness theorem of sentential logic, some ﬁnite subset of is unsatisﬁable. This fact can be used to obtain a proof of the compactness theorem for ﬁrst-order logic, not relying on Section 2.5 or the deductive calculus of Section 2.4. Moreover, a proof of the enumerability theorem, similarly independent of Sections 2.4 and 2.5, can be extracted from the Herbrand approach. Take the special case = ∅. If ϕ is valid then as we produce

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A Mathematical Introduction to Logic more and more of , at some point we will have an unsatisﬁable set, a fact we can recognize by use of truth-tables. If ϕ is not valid, then as we produce more and more of , we are making a structure in which ϕ fails, but the structure is inﬁnite and the construction never terminates.

Exercises 1. Prove the L¨owenheim–Skolem theorem in the following improved form: Let A be a structure for a countable language. Let S be a countable subset of |A|. Then there is a countable substructure B of A with S ⊆ |B| with the property that for any function s mapping the variables into |B| and any (ﬁrst-order) ϕ, |=A ϕ[s]

2.

3.

4.

5.

6. 7.

iff

|=B ϕ[s].

Suggestion: Choose Skolem functions for all formulas. Close S under the functions. Remark: A substructure B with this property is said to be an elementary substructure. Note that the property implies (by taking ϕ to be a sentence) that A ≡ B. On the one hand, this form gives a stronger conclusion than we had in Section 2.6. Not only do we get that Th A has some countable model, we get a countable submodel. On the other hand, the proof uses the axiom of choice. Extend the previous exercise to the uncountable case. Assume that A is a structure for a language of cardinality λ. Let S be a subset of |A| having cardinality κ. Show that there is an elementary substructure B of A of cardinality at most κ + λ with S ⊆ |B|. Show that Corollary 42B is optimal in the following sense: (a) Given any ∃1 sentence σ we can effectively decide whether or not σ is satisﬁable. (b) Given any ∀1 sentence σ we can effectively decide whether or not σ is valid. (a) State the two corollaries (analogous to 42A and 42B) described at the end of this section. (b) Supply proofs. Repeat the example given for Herbrand expansions, but for the converse: ∃ y ∀ x P x y |= ∀ x ∃ y P x y. Show that in this case the set is unsatisﬁable in sentential logic. Apply the method of Herbrand expansions to establish the following: |= ∃ x(P x → ∀ x P x). Modify the Herbrand expansion construction to accomodate a language with equality. Suggestion: In effect, step 5 of the completeness proof in Section 2.5 must be added. Add enough universal sentences to assure that { t1 , t2 | v(t1 = t2 ) = T is a congruence relation.

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SECTION 4.3 Many-Sorted Logic We now return to ﬁrst-order languages, but with many sorts of variables, ranging over different universes. (In the next section this will be applied to the case in which one sort of variable is for elements of a universe, another for subsets of that universe, yet another for binary relations, and so forth.) In informal mathematics one sometimes says things like, “We use Greek letters for ordinals, capital script letters for sets of integers, . . . .” In effect, one thereby adopts several sorts of variables, each sort having its own universe. We now undertake to examine this situation precisely. As might be expected, nothing is drastically different from the usual one-sorted situation. None of the results of this section are at all deep, and most of the proofs are omitted. Assume that we have a nonempty set I , whose members are called sorts, and symbols arranged as follows: A. Logical symbols 0. Parentheses: (, ). 1. Sentential connective symbols: ¬, →. 2. Variables: For each sort i, there are variables v1i , v2i , . . . of sort i. 3. Equality symbols: For some i ∈ I there may be the symbol =i , said to be a predicate symbol of sort i, i . B. Parameters 0. Quantiﬁer symbols: For each sort i there is a universal quantiﬁer symbol ∀i . 1. Predicate symbols: For each n > 0 and each n-tuple i 1 , . . . , i n of sorts, there is a set (possibly empty) of n-place predicate symbols, each of which is said to be of sort i 1 , . . . , i n . 2. Constant symbols: For each sort i there is a set (possibly empty) of constant symbols each of which is said to be of sort i. 3. Function symbols: For each n > 0 and each (n + 1)-tuple i 1 , . . . , i n , i n+1 of sorts, there is a set (possibly empty) of n-place function symbols, each of which is said to be of sort i 1 , . . . , i n , i n+1 . As usual, we must assume that these categories of symbols are disjoint, and further that no symbol is a ﬁnite sequence of other symbols. Each term will be assigned a unique sort. We deﬁne the set of terms of sort i inductively, simultaneously for all i:

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A Mathematical Introduction to Logic 1. Any variable of sort i or constant symbol of sort i is a term of sort i. 2. If t1 , . . . , tn are terms of sort i 1 , . . . , i n , respectively, and f is a function symbol of sort i 1 , . . . , i n , i n+1 then f t1 · · · tn is a term of sort i n+1 . This deﬁnition can be recast into a more familiar form. The set of pairs t, i such that t is a term of sort i is built up from (i.e., generated from) the basic set { vni , i | n ≥ 1 & i ∈ I } ∪ { c, i | c is a constant symbol of sort i} by the operations that, for a function symbol f of sort i 1 , . . . , i n , i n+1 , produce the pair f t1 · · · tn , i n+1 from the pairs t1 , i 1 , . . . , tn , i n . An atomic formula is a sequence Pt1 · · · tn consisting of a predicate symbol of sort i 1 , . . . , i n and terms t1 , . . . , tn of sort i 1 , . . . , i n , respectively. The nonatomic formulas are then formed using the connectives ¬, → and the quantiﬁers ∀i vni . A many-sorted structure A is a function on the set of parameters which assigns to each the correct type of object: 1. To the quantiﬁer symbol ∀i , A assigns a nonempty set |A|i called the universe of A of sort i. 2. To each predicate symbol P of sort i 1 , . . . , i n , A assigns a relation P A ⊆ |A|i1 × · · · × |A|in . 3. To each constant symbol c of sort i, A assigns a point cA in |A|i . 4. To each function symbol f of sort i 1 , . . . , i n , i n+1 , A assigns a function f A : |A|i1 × · · · × |A|in → |A|in+1 . The deﬁnitions of truth and satisfaction are the obvious ones, given that ∀i is to mean “for all members of the universe |A|i of sort i.” In a many-sorted structure, the universes of the various sorts might or might not be disjoint. But since we have no equality symbols between sorts, any nondisjointness must be regarded as accidental. In particular, there will always be an elementarily equivalent structure whose universes are disjoint.

Reduction to One-Sorted Logic Many-sorted languages may at times be convenient (as in the following section). But there is nothing essential that can be done with them that cannot already be done without them. We now proceed to make this assertion in a more precise form.

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We will consider a one-sorted language having all the predicate, constant, and function symbols of our assumed many-sorted language. In addition, it will have a one-place predicate symbol Q i for each i in I . There is a syntactical translation taking each many-sorted formula ϕ into a one-sorted formula ϕ ∗ . In this translation all equality symbols are replaced by =. The only other change is in the quantiﬁers (the quantiﬁer symbols and the quantiﬁed variables): We replace ∀i vni

vni

by ∀ v(Q i v →

v

),

where v is a variable chosen not to conﬂict with the others. Thus the quantiﬁers of sort i are “relativized” to Q i . (The free variables are left alone.) Turning now to semantics, we can convert a many-sorted structure A into a structureA∗ for the above one-sorted language. The universe |A∗ | is the union i∈I |A|i of all the universes of A. To Q i is assigned the set |A|i . On the predicate and constant symbols, A∗ agrees with A. ∗ For a function symbol f , the function f A is an arbitrary extension of ∗ f A . (Of course this last sentence does not completely specify f A . The results we give for A∗ hold for any structure obtained in the manner just described.) LEMMA 43A A many-sorted sentence σ is true in A iff σ ∗ is true in A∗ . To prove this, one makes a stronger statement concerning formulas: |=A ϕ[s] ⇐⇒ |=A∗ ϕ ∗ [s] where s(vni ) ∈ |A|i . The stronger statement is then proved by induction. Consider now the other direction. A one-sorted structure is not always convertible into a many-sorted structure. So we will impose some conditions. Let be the set consisting of the following one-sorted sentences: 1. ∃ v Q i v, for each i in I . 2. ∀ v1 · · · ∀ vn (Q i v1 → · · · → Q in vn → Q in+1 f v1 · · · vn ), for each function symbol f of sort i 1 , . . . , i n , i n+1 . We include the case n = 0, in which case the above becomes the sentence Q i c for a constant symbol c of sort i. Notice that the above A∗ was a model of . A one-sorted model B of does convert into a many-sorted B . The conversion is performed

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A Mathematical Introduction to Logic in the natural way: |B |i = Q iB ; P B = P B ∩ (Q iB1 × · · · × Q iBn ), where P is a predicate symbol of sort i 1 , . . . , i n cB = cB ; f B = f B ∩ (Q iB1 × · · · × Q iBn × Q iBn+1 ), the restriction of f B to Q iB1 × · · · × Q iBn , where f is a function symbol of sort i 1 , . . . , i n , i n+1 . LEMMA 43B If B is a model of , then B is a many-sorted structure. Furthermore, a many-sorted sentence σ is true in B iff σ ∗ is true in B. The proof is similar to the proof of Lemma 43A. Notice that B∗ is not in general equal to B. (For example, |B| may contain points not belonging to any Q iB .) On the other hand, A∗ is equal to A. THEOREM 43C In the many-sorted language |= σ iff in the one-sorted language ∗ ∪ |= σ ∗ . PROOF. (⇒) Assume that |= σ and let B be a one-sorted model of ∗ ∪ (where ∗ = {σ ∗ | σ ∈ }). Then B is a model of by Lemma 43B. Hence B is a model of σ . So by Lemma 43B again, B is a model of σ ∗ . (⇐) Similar, with Lemma 43A. By using Theorem 43C, we can now infer the following three theorems from the corresponding one-sorted results. COMPACTNESS THEOREM If every ﬁnite subset of a set of manysorted sentences has a model, then has a model. PROOF. Assume that every ﬁnite subset 0 of has a many-sorted model A0 . Then a ﬁnite subset 0∗ of ∗ has the model A∗0 . Hence by the ordinary compactness theorem, ∗ has a model B. B is then a model of . ENUMERABILITY THEOREM For a recursively numbered many-sorted language, the set of G¨odel numbers of valid sentences is recursively enumerable. PROOF.

For a many-sorted σ , we have by Theorem 43C, |= σ

iff

|= σ ∗ .

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Since is recursive, Cn is recursively enumerable. And σ ∗ depends recursively on σ , so we can apply Exercise 7(b) of Section 3.5. LOWENHEIM ¨ –SKOLEM THEOREM For any many-sorted structure (for a countable language) there is an elementarily equivalent countable structure. PROOF. Say that the given structure is A. Then A∗ is a one-sorted model of (Th A)∗ ∪. Hence by the ordinary L¨owenheim–Skolem theorem, (Th A)∗ ∪ has a countable model B. B is a model of Th A and so is elementarily equivalent to A.

SECTION 4.4 General Structures We now return to the discussion of second-order logic begun in Section 4.1. We discussed there (a) the syntax, i.e., the set of wffs for second order, and (b) the semantics, i.e., the concept of structure (which was the same as for ﬁrst order) and the deﬁnition of satisfaction and truth. In this section we want to leave (a) unchanged, but we want to present an alternative to (b). The idea can be stated very brieﬂy: We view the language (previously thought of as second-order) now as being a manysorted elementary (i.e., ﬁrst-order) language. The result is to make open to interpretation not only the universe over which individual variables range but also the universes for the predicate and function variables. This approach is particularly suited to number theory; that case is examined brieﬂy at the end of this section.

The Many-Sorted Language Despite the fact that we want ultimately to consider the grammar of Section 4.1, it will be expedient to consider also a many-sorted (ﬁrst-order) language constructed from the second-order language of Section 4.1. We take ℵ0 sorts: the one individual sort (with variables v1 , v2 , . . .); for each n > 0, the n-place predicate sort (with variables Xn1 , Xn2 , . . .); and for each n > 0, the n-place function sort (with variables Fn1 , Fn2 , . . .). We will use equality (=) only between terms of the individual sort. The predicate and function parameters of our assumed second-order language will also be parameters of the many-sorted language, and will take as arguments terms of the individual sort. (For a function parameter f , the term f t is of the individual sort. The only terms of predicate or function sort are the variables of those sorts.) In addition, we now use two new classes of parameters. For each n > 0 there is a membership predicate parameter εn which takes as

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A Mathematical Introduction to Logic arguments one term of the n-place predicate sort (i.e., a variable Xnm ) and n terms of the individual sort. Thus, for example, ε3 X3 v2 v1 v8 is a wff. Its intended interpretation is that the triple denoted by v2 , v1 , v8 is to belong to the relation denoted by X3 . This is exactly the interpretation assigned previously to the second-order formula X 3 v2 v1 v8 , and, in fact, readers are advised to identify these two formulas closely in their minds. For each n > 0, there is also the evaluation function parameter E n . E n takes as arguments one term of the n-place function sort (i.e., a variable Fnm ) and n terms of the individual sort. The resulting term, E n Fn t1 · · · tn , is itself of the individual sort. Again readers are advised to identify closely the term E n Fn t1 · · · tn with the previous Fn t1 · · · tn . There is an obvious way of translating between the second-order language of Section 4.1 and the present many-sorted language. In one direction we stick on the εn and E n symbols; in the other direction we take them off. The purpose of these symbols is to make the language conform to Section 4.3. A many-sorted structure has universes for each sort and assigns suitable objects to the various parameters (as described in the preceding section). First, we want to show that without loss of generality, we may suppose that εn is interpreted as genuine membership and E n as genuine evaluation. THEOREM 44A Let A be a structure for the above many-sorted language such that the different universes of A are disjoint. Then there is a homomorphism h of A onto a structure B such that (a) h is one-to-one, in fact the identity, on the individual universe (from which it follows that |=A ϕ[s]

iff

|=B ϕ[h ◦ s]

for each formula ϕ). (b) The n-place predicate universe of B consists of certain nary relations over the individual universe, and R, a1 , . . . , an is in εnB iff a1 , . . . , an ∈ R. (c) The n-place function universe of B consists of certain n-place functions on the individual universe, and E nB ( f, a1 , . . . , an ) = f (a1 , . . . , an ). PROOF. Since the universes of A are disjoint, we can deﬁne h on one universe at a time. On the individual universe U , h is the identity.

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On the universe of the n-place predicate sort, h(Q) = { a1 , . . . , an | each ai is in U and Q, a1 , . . . , an is in εnA }. Thus a1 , . . . , an ∈ h(Q)

iff

Q, a1 , . . . , an is in εnA .

(1)

Similarly, on the universe of the n-place function sort, h(g) is the n-place function on U whose value at a1 , . . . , an is E nA (g, a1 , . . . , an ). Thus h(g)(a1 , . . . , an ) = E nA (g, a1 , . . . , an ).

(2)

For εnB we take simply the membership relation, R, a1 , . . . , an is in εnB

iff

a1 , . . . , an ∈ R.

(3)

For E nB we take the evaluation function, E nB ( f, a1 , . . . , an ) = f (a1 , . . . , an ).

(4)

On the other parameters (inherited from the second-order language) B agrees with A. Then it is clear upon reﬂection that h is a homomorphism of A onto B. That h preserves εn follows from (1) and (3), where in (3) we take R = h(Q). Similarly, from (2) and (4) it follows that h preserves E n . Finally, we have to verify the parenthetical remark of part (a). This follows from the many-sorted analogue of the homomorphism theorem of Section 2.2, by using the fact that we have equality only for the individual sort, where h is one-to-one. By the above theorem, we can restrict attention to structures B in which εn and E n are ﬁxed by (b) and (c) of the theorem. But since εnB and E nB are determined by the rest of B, we really do not need them at all. When we discard them, we have a general pre-structure for our second-order grammar.

General Structures for Second-Order Languages These structures provide the alternative semantics mentioned at the beginning of this section. DEFINITION. A general pre-structure A for our second-order language consists of a structure (in the original sense), together with the additional sets: (a) For each n > 0, an n-place relation universe, which is a set of n-ary relations on |A|;

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A Mathematical Introduction to Logic (b) For each n > 0, an n-place function universe, which is a set of functions from |A|n into |A|. A is a general structure if, in addition, all comprehension sentences are true in A. The last sentence of the deﬁnition requires explanation. First, a comprehension sentence is a sentence obtained as a generalization of a comprehension formula (see Example 3 of Section 4.1). Thus it is a generalization of ∃ Xn ∀ v1 · · · ∀ vn (Xn v1 · · · vn ↔ ϕ), where Xn does not occur free in ϕ, or a generalization of ∀ v1 · · · ∀ vn ∃!vn+1 ψ→ ∃ Fn ∀ v1 · · · ∀ vn+1 (Fn v1 · · · vn = vn+1 ↔ ψ), where Fn does not occur free in ψ. (Here ϕ and ψ can have individual variables, predicate variables, and function variables.) Next we must say what it means for a comprehension sentence (or any second-order sentence for that matter) to be true in A. Assume then that A is a general pre-structure. Then a sentence σ is true in A iff the result of converting σ into a many-sorted sentence (by adding εn and E n ) is true in A, with εn interpreted as membership and E n as evaluation. More generally, let ϕ be a second-order formula, and let s be a function that assigns to each individual variable a member of |A|, to each predicate variable a member of the relation universe of A, and to each function variable a member of the function universe of A. Then we say that A satisﬁes ϕ with s (written |=G A ϕ[s]) iff the many-sorted version of ϕ is satisﬁed with s in the structure A, where εn is interpreted as membership and E n as evaluation. The essential consequences of this deﬁnition of satisfaction are the following, which should be compared with 5 and 6 of page 283. n |=G A ∀ X ϕ[s] n |=G A ∀ F ϕ[s]

iff for every R in the n-place relation n universe of A, |=G A ϕ[s(X | R)]. iff for every f in the n-place relation n universe of A, |=G A ϕ[s(F | f )].

This, then, is the alternative approach mentioned at the beginning of the section. It involves treating the second-order grammar as being a many-sorted ﬁrst-order grammar in disguise. Because this approach is basically ﬁrst-order, we have the L¨owenheim–Skolem theorem, the compactness theorem, and the enumerability theorem. LOWENHEIM ¨ –SKOLEM THEOREM If the set of sentences in a countable second-order language has a general model, then it has a countable general model.

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Here a countable general model is one in which every universe is countable (or equivalently, the union of all the universes is countable). PROOF. Let be the set of comprehension sentences. Then ∪ , viewed as a set of many-sorted sentences, has a countable manysorted model by the L¨owenheim–Skolem theorem of the preceding section. By Theorem 44A, a homomorphic image of that model is a general pre-structure satisfying ∪ , and hence is a general model of . COMPACTNESS THEOREM If every ﬁnite subset of a set of secondorder sentences has a general model, then has a general model. PROOF. The proof is exactly as above. Every ﬁnite subset of ∪ has a many-sorted model, so we can apply the compactness theorem of the preceding section. ENUMERABILITY THEOREM Assume that the language is recursively numbered. Then the set of G¨odel numbers of second-order sentences that are true in every general structure is recursively enumerable. PROOF. A sentence σ is true in every general structure iff it is a many-sorted consequence of . And is recursive. The above two theorems assure us that there is an acceptable deductive calculus such that τ is deducible from iff τ is true in every general model of (see the remarks at the beginning of Section 2.4). But now that we know there is such a complete deductive calculus, there is no compelling reason to go into the detailed development of one. We can compare the two approaches to second-order semantics as follows: The version of Section 4.1 (which we will now call absolute second-order logic) is a hybrid creature, in which the meaning of the parameters is left open to interpretation by structures, but the concept of being (for example) a subset is not left open, but is treated as having a ﬁxed meaning. The version of the present section (general second-order logic) avoids appealing to a ﬁxed notion of subset, and consequently is reducible to ﬁrst-order logic. In this respect it is like axiomatic set theory, where one speaks of sets and sets of sets and so forth, but the theory is a ﬁrst-order theory. By enlarging the class of structures, general second-order logic diminishes the cases in which logical implication holds. That is, if every general model of is a general model of σ , it then follows that |= σ in absolute second-order logic. But the converse fails. For example, take = ∅: The set of sentences true in all general models is a recursively enumerable subset of the nonarithmetical set of valid sentences of absolute second-order logic.

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Models of Analysis We can illustrate the ideas of this section by focusing attention on the most interesting special case: general models of second-order number theory. Consider then the second-order language for number theory, with the parameters 0, S, <, ·, and E. We take as our set of axioms the set A2E obtained from A E by adding as a twelfth member the Peano induction postulate (Example 2, Section 4.1). From Exercise 1 of Section 4.1, we can conclude that any model (in the semantics of that section) of A2E is isomorphic to N. But what of the general models of our axiom set? They can differ from N in either (or both) of two ways. We can employ the compactness theorem as before to construct (nonstandard) general models of the axioms having inﬁnite numbers (i.e., models A with an element larger than the denotation of Sn 0 in the ordering

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THEOREM 44B If A and B are ω-models of analysis having the same one-place relation universe, then A = B. PROOF. Suppose R belongs to the three-place relation universe of A. Let R be the “compression” of R into a unary relation:

R = { a, b, c | a, b, c ∈ R}. Our sequence-encoding function is recursive and hence is ﬁrstorder deﬁnable in number theory by a formula ϕ. R is in the set universe of A by virtue of the comprehension sentence ∀ X3 ∃ X1 ∀ u[X1 u ↔ ∃ v1 ∃ v2 ∃ v3 (ϕ(v1 , v2 , v3 , u) ∧ X3 v1 v2 v3 )]. Thus R is in the set universe of B; we unpack it by a similar argument. R is in the three-place relation universe of B by virtue of the comprehension sentence ∀ X1 ∃ X3 ∀ v1 ∀ v2 ∀ v3 [X3 v1 v2 v3 ↔ ∃ u(ϕ(v1 , v2 , v3 , u) ∧ X1 u)]. A similar argument applies to the function universes.

Consequently, we can identify an ω-model of analysis with its set universe (which is included in PN). Not every subclass of PN is then an ω-model of analysis, but only those for which the comprehension sentences are satisﬁed. EXAMPLES OF ω-MODELS.

We need only specify the set universe.

1. PN is the absolute model. 2. Let (A; ∈ A ) be a model of the usual axioms for set theory such that (i) the relation ∈ A is the genuine membership relation { a, b | a ∈ A, b ∈ A, and a ∈ b} on the universe A, and (ii) A is transitive, i.e., if a ∈ b ∈ A, then a ∈ A. Then the collection of all those subsets of N that belong to A is an ω-model of analysis. 3. For a class A ⊆ PN, deﬁne DA to be the class of all sets B ⊆ N which are deﬁnable in the ω pre-structure with set universe A by a formula of the language of second-order number theory, augmented by parameters for each set in A. Then deﬁne by transﬁnite recursion on the ordinals: A0 = ∅, Aα+1 = DA α, Aλ = α<λ Aα

for limit λ.

By cardinality considerations we see that this stops growing at some ordinal β for which Aβ+1 = Aβ . Let β0 be the least

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A Mathematical Introduction to Logic such β; it can be shown (from the L¨owenheim–Skolem theorem) that β0 is a countable ordinal. Aβ0 coincides with α Aα (the union being over all ordinals α) and is called the class of ramiﬁed analytical sets. It is an ω-model of analysis; the truth of the comprehension sentences follows from the fact that DAβ0 ⊆ Aβ0 .

Suggestions for Further Reading Jon Barwise (editor). Handbook of Mathematical Logic. North-Holland Publishing Company, Amsterdam, 1978. This “handbook” collects 31 expository articles on model theory, set theory, recursion theory, and proof theory, written by experts. Jon Barwise and John Etchemendy. The Language of First-Order Logic. Center for the Study of Language and Information, Stanford, 1992. This introductory textbook comes with a disk for the Tarski’s World software package. The same authors have produced the Turing’s World and Hyperproof software packages. J. L. Bell and M. Machover. A Course in Mathematical Logic. NorthHolland Publishing Company, Amsterdam, 1977. George Boolos and Richard Jeffrey. Computability and Logic. Cambridge University Press, Cambridge, 1974 (third edition 1989). This textbook gives a lively treatment of some of topics in the present book, addressed to a different audience. C. C. Chang and H. J. Keisler. Model Theory. North-Holland Publishing Company, Amsterdam, 1973 (third edition 1990). This remains the classic book on model theory. Herbert B. Enderton. Elements of Set Theory. Academic Press, New York, 1977. This is the author’s favorite book on set theory. Wilfrid Hodges. A Shorter Model Theory. Cambridge University Press, Cambridge, 1997. This is a shorter version of his Model Theory, published in 1993. Hartley Rogers. Theory of Recursive Functions and Effective Computability. McGraw-Hill Book Company, New York, 1967. This book is still the standard work in its ﬁeld. Joseph R. Shoenﬁeld. Mathematical Logic. Association for Symbolic Logic and A K Peters, Natick, Massachusetts, 2000. First published by Addison-Wesley in 1967, the book gives a compact graduate-level treatment.

307

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A Mathematical Introduction to Logic Jean van Heijenoort (editor). From Frege to G¨odel: A Source Book in Mathematical Logic, 1879–1931. Harvard University Press, Cambridge, Massachusetts, 1967. This is a collection of 46 fundamental papers in logic, translated into English and supplied with commentaries.

List of Symbols The numbers indicate the pages on which the symbol ﬁrst occurs. ∈ ∈ / = A; t ∅ {x1 , . . . , xn } {x| x } N Z ⊆ P ∪ ∩ x1 , . . . , xn A×B dom R ran R ﬂd R An F:A→B [X ] A∼B card A ℵ R

1 1 1 1 2 2 2 2 2 2 2 3 3 3 3 4 4 4 4 4 5 5 5 8 8 8 9 9

⇒ ⇐ ⇔ ∴

|= ¬ → ∧ ∨ ↔ E F T v |= |== D # Bαn ⊥ ↓ | +

∀ ∃

1 1 1 1 1 1 11 11 11 12 14 17 20 20 20 1 24 32 45 46 50 50 51 51 51 62 68 68, 77

vn = 0 S < + · E Ff Qi = < |A| c21 |=A ρ[s] § s(x|d) |= |==| Mod EC EC |=A ϕ[[a1 , . . . , ak ]] A∼ =B h◦s A≡B Q ∀n

68 69 70, 182 71, 182 70, 182 71, 182 71, 182 71, 182 74 75 1, 78 78 81 81 83 83 84 88 88 92 92 92 90 94 96 97 98 00

309

310 ∃n ∃! ! αtx

! Q2a Eqn Ax T gen MP ded RAA Th Cn A ST ϕ(t1 , . . . , tn ) πS π B π −1 [T1 ] ϕπ ∗ A F I # st R see pp. 182–192

List of Symbols 00 102 110 110 112 121 113 114 122 122 122 122 122 122 152 155 161 167 168 168 168 169 00 175 176 177 178 182

Sk 0 α G S4n AS AL ≤

≤ Ln ∧i ≡n ≡n ∨i AE An Mn En Iim μb− KR pn a0 , . . . , am (a)b lh ab f a∗b ∗

183 184 184 188 188 194 194 194 194 195 197 197 00 203 203 203 203 214 216 217 219 220 220 221 221 221 222 223

∃ ∀ Sb n

225 225 228 230 242

n ST AM Tm U [[e]]m K We ρ Xin Fin Qi A∗ B εn En |=G A ϕ[s] R DA

242 270 276 249 249 252 254 255 258 282 283 297 297 297 297 299 300 302 305 305

Index A Abbreviations, 1 Absolute model, 304, 305 Absolute second-order logic, 303 Adjoining, 2 Algebraically closed ﬁelds, 158–159 Algebraic numbers, 10 Algorithm, 61 Alphabetic variants, 126–127 Analysis, models of, 304–306 Analysis, nonstandard. See Nonstandard analysis Arithmetic. See Number theory Arithmetical hierarchy, 242–245 Arithmetical relations, 100, 242 Arithmetization of syntax, 224–234 Asser, G¨unter, 101 Atomic formulas, 74–75, 83 Automorphism, 98–99 Axiomatizable theory, 156–157 Axioms, logical. See Logical axioms

B Berkeley, George, 173 Biconditional symbol, 14 Binary connectives, 51

Bolzano–Weierstrass theorem, 181 Boolean algebra, 20 Boolean functions, 45–52 Bounded quantiﬁers, 204, 210–211 Bound variables, 80 Bridge circuit, 57

C Calculus, deductive. See Deductive calculus Cantor, Georg, 8 Cantor’s theorem, 159, 163 Capital asterisk operation, 223 Cardinal arithmetic theorem, 9–10 Cardinality of languages, 141 Cardinality of structures, 153–154, 157 Cardinal numbers, 8–10 Carroll, Lewis, 162 Cartesian product, 4 Categorical sets, 154, 157 Categoricity in power, 157 Chain, 7 Chain rule, 180 Characteristic function, 217 Chinese remainder theorem, 91, 279 Church’s theorem, 145, 164, 238 Church’s thesis, 185, 187, 206–210, 233–234, 240, 247

Circuits, switching, 54–59 Closed, 5, 18, 35, 111 Compactness theorem history of, 145 in ﬁrst-order logic, 109, 142, 293 in many-sorted logic, 298 in second-order logic, 285, 303 in sentential logic, 24, 59–60 Completeness theorem, 66, 135–145 Complete sets of connectives, 49 Complete theory, 156 Composition, 5, 215–216 Comprehension formulas, 284 Computability approach to incompleteness, 184, 187, 257–258 Computable, 65, 208–209 Computable functions, 209–210, 250–251 See also Recursive functions Computably enumerable (c.e.), 238 Computing agents, idealized, 208, 261–263 Concatenation function, 222–223 Conditional sentence, 21 Conditional symbol, 14 Congruence relation, 140 Conjunction symbol, 14

311

312

Index

Conjunctive normal form (CNF), 53 Connectives. See Sentential connectives Consequences, set of, 155 Consequent, 113 Consistent sets, 119, 135 Constants, generalization on, 123–124 Constant symbols, 70, 79 Construction sequences, 17–18, 35–37, 111 Contraposition, 27, 119, 121 Convergence, 178–180 Countable language, 135, 145, 151–153 Countable sets, 6 C++, 13 Craig’s theorem, 163

D D’Alembert, Jean, 173 Decidable sets, 62–63, 144, 185 See also Church’s thesis Decidable theory, 144, 157 See also Undecidability Decoding function, 220 Deducible formulas, 111 Deductions, 66, 110–112 Deduction theorem, 118–120 Deductive calculus, 66, 109 alphabetic variants, 126–127 equality, 127–128 formal deductions, 110–112 metatheorems and, 116–120 strategy, 120–126 substitution, 112–114 tautologies, 114–116 Deﬁnability in a structure, 90–92 of a class of structures, 92–94 Deﬁnable element, 91 Deﬁnable relations, 90–92, 98, 287 from points, 103

Deﬁned function symbols, 164–166, 169, 172 Deﬁnition by recursion, 38–44 Delay of circuit, 56 De Morgan’s laws, 27, 49 Dense order, 159 Depth of circuit, 56 Derivability conditions, 267 Descriptions. See Deﬁned function symbols Diagonal function, 264 Diagonalization approach to incompleteness, 184, 186–187, 245–246 Directed graphs (digraphs), 82, 93 Disjoint set, 3 Disjunction symbol, 14 Disjunctive normal form (DNF), 49 Divisibility, 218 Domain of relation, 4 of structure, 81 Dominance, 8–9 Donkey sentences, 80 Double negation, 89 Dovetailing, 64 Duality, 28

Entscheidungsproblem, 164 Enumerability theorem, 109, 142–143, 145, 293 in many-sorted logic, 298 in second-order logic, 286, 303 Equality, 1–2, 127–128 language of, 246, 285 Equality symbol, 70 Equinumerous, 8 Equivalence classes and relations, 6, 189 Euler, Leonhard, 5, 173 Evaluation function parameter, 300 Eventually periodic set, 201 Excluded middle, 27 Exclusive disjunction, 51 Existential formula (∃1 ), 102, 205 Existential instantiation (rule EI), 124–125, 145 Existential quantiﬁers, 67, 87, 287, 288 Exponential growth, 26 Exponentiation, representation of, 276–281 Exportation, 27 Expressions, 15–16, 73–74 Extension, 95 Extensionality, principle of, 2

E Effective computability, 65 See also Recursive functions Effective enumerability, 63–66 See also Recursively enumerable relations Effective procedures, 61–65 See also Church’s thesis Elementarily closed (ECL), 104 Elementary class (EC, EC ), 92–93 Elementary equivalence, 97 Elementary substructure, 294 Elementary type, 104 Eliminable deﬁnition, 172 Elimination of quantiﬁers, 190–192

F Faithful interpretations, 171–172 Falsity, 20 Field (of relation), 4 Fields, 87, 92, 93–94, 285 real-closed, 104 theory of, 155–156, 158–159 See also Algebraically closed ﬁelds Finite graphs, 93 Finite language, 142 Finitely axiomatizable theories, 156 Finitely valid, 147 Finite model property, 163 Finite models, 147–151

Index Finite sequence (string), 4 Finite set, 6 First-order language, 67–72, 167 examples of, 70–73 formulas, 73–76 free variables, 76–77 notation, 77–79 First-order logic completeness theorem, 135–145 deductive calculus, 109–129 interpretations between theories, 164–172 language of, 69–79 models of theories, 147–162 parsing algorithm, 105–108 soundness theorem, 131–135 translation methods, 68–69 truth and models, 80–99 Fischer, Michael, 201 Fixed-point lemma, 234–235 Formal languages, 11–13 computer, 13 features in, 11–13 sentential logic and, 13–19 Formula-building operations, 17, 75 Formulas atomic, 74–75, 83 comprehension, 284 generalization of, 116 satisfaction of, 83–86 unique readability of, 40–41, 108 well-formed (wffs), 12, 17–18, 75 Freely generated sets, 39–40 See also Unique readability theorem Free variables, 76–77 Frege, Gottlob, 152 Function comprehension formulas, 284 Functions, 5 deﬁning, 164–166 recursive, 247–263 representable, 212–217 Skolem, 145, 287–290

313 Function symbols, 70, 79, 128 Function universe, 302 Function variables, 282

Homomorphism theorem, 96–97 Hyperreal numbers. See Nonstandard analysis Hypothesis, 23, 67, 109, 213

G I Generalization on constants, 123–124 of formulas, 112 Generalization theorem, 117–118 General pre-structure, 301 General second-order logic, 303 General structures, 299–306 Generated sets, 37 freely, 39 G¨odel, Kurt, 145, 152 β-function, 278–279, 281 completeness theorem, 135–145 incompleteness theorem, 145, 236, 256, 257–258 numbers, 91, 184, 225–234, 286 second incompleteness theorem, 266–270, 274–275 Goldbach’s conjecture, 263 Graphs, 92 connected, 146 directed, 82, 93 ﬁnite, 93 of function, 209 Groups, 38, 92

H Halting problem, unsolvability of, 254 Henkin, Leon, 145 Herbrand expansions, 290–294 Herbrand, Jacques, 293 Herbrand’s theorem, 293 Herbrand universe, 291 Heterological, 186 Hilbert, David, 152 Homomorphisms, 94–99

Identity function, 5 Identity interpretation, 168 Iff, use of, 1 Implicant, 59 Implicitly deﬁnable relations, 287 Incompleteness theorem (G¨odel) ﬁrst, 145, 236, 256, 257–258 second, 266–270, 274–275 undecidability and, 234–245 Inconsistent sets, 119 Inconsistent sets. See Consistent sets Independent axiomatizations, 28 Index of recursively enumerable set, 255 of recursive partial function, 253 Individual variables, 282–283 Induction, 30, 34–38 principle, 18–19, 37, 44, 111–112 Inductive sets, 35 Induction axiom. See Peano induction postulate Inﬁnitely close, 177 Inﬁnitesimal, 176 Initial segment, 4 Input/output format, 62, 209 Instances, 291 Integers, 2 Interpolation theorem, 53 Interpretations, 80 between theories, 164–172, 273 Intersection, 3 Isomorphic embedding, 94 Isomorphic structures, 94 Isomorphism, 94

314

Index K

Kleene normal form, 249–250, 252–254, 257 Kleene’s theorem, 64, 239

L Lagrange’s theorem, 166 Languages many-sorted, 299–301 of equality, 285 See also First-order languages, Formal languages, Second-order logic Least-zero operator, 216, 220–221 Leibniz, G. W. v., 173 Length, 221 Lindenbaum’s theorem, 246 Linear connectives, 52 Linear transformations, 99 Literal, 59 L¨ob’s theorem, 269 Logical axioms, 110, 112, 125 recursiveness of, 232 validity of, 131–134 Logical implication, 88–99 Logically equivalence, 88 Logical symbols, 14, 69–70 Ło´s–Vaught test, 157–160, 190 L¨owenheim, Leopold, 151 L¨owenheim–Skolem theorem, 103, 151–155, 190 in many-sorted logic, 299 in second-order logic, 285, 302–303 LST theorem, 154 Łukasiewicz, Jan, 33 See also Polish notation

M Majority connective, 45 Mal´cev, Anatolii, 145 Many-one reducibility, 256

Many-sorted logic, 295–299 application to second-order logic, 299–301 Many-valued logic, 20 Map, 5 Map coloring, 65, 146 Material conditional, 21 Membership predicate, 299–300 Meta-language, 89, 129 Metamathematics, use of term, 69 Metatheorems, 116–120 Models, 80–99 of analysis, 304–306 of theories, 147–162 Modus ponens, 66, 110–111, 116 Monotone connectives, 54 Monotone recursion, 224 μ-operator, 216, 220–221

N Nand, 51 Natural numbers, 2 See also Number theory Negation symbol, 14, 17 Newton, Isaac, 173 Nonlogical symbols, 14 Nonprime formulas, 114 Nonstandard analysis, 173–181 algebraic properties, 176–178 construction of hyperreals, 173–176 convergence in, 178–180 Nonstandard models, 152–153, 183, 304 Normal form theorem, for recursive functions, 252–253 Skolem, 288–289 Notation, 77–79 NP, 26, 101 Number theory, 182 language of, 70, 72, 182 with addition, 196–197, 280 with exponentiation, 202–205, 280

with multiplication, 276–281 with ordering, 193–196, 280 with successor, 187–193, 280 Numerals, 183–184, 209 Numeralwise determined formulas, 206, 210–212

O Object language, 89 Occur free, 76–77 ω-completeness, 223 ω-consistency, 241, 245 ω-models of analysis, 304–306 One-sorted logic, 296–299 One-to-one functions, 5 Onto, 5 Operating system, 253 Operations, 5 Ordered n-tuples, 3–4 Ordered pairs, 3, 4 Ordering relations, 6, 93, 159, 284

P Pairing function, 220, 277–278 Pairwise disjoint set, 3 Parameters, 14, 70 Parameter theorem, 258–260, 264 Parentheses, use of, 33, 78 Parity connective, 53 Parsing algorithm in ﬁrst-order logic, 105–108 in sentential logic, 29–33 Parsing formulas, 29–33, 107–108 Parsing terms, 106–107 Partial functions, 250 Partial recursive functions. See Recursive functions, partial Partition, 6

Index

315

Peano arithmetic (PA), 269–270 Peano induction postulate, 193, 284, 286–287 Periodic set, 201 Permutation, 100 Polish notation, 32–33, 74 Polynomial-time decidable, 26, 115 Post, Emil, 47, 152, 261 Power set, 2–3 Predicate calculus. See First-order logic Predicate symbols, 70, 79, 128 Predicate variables, 282 Prenex formulas, 160 Prenex normal form, 160–161 Presburger’s theorem, 197–198 Prime formulas, 114–115 Prime implicants, 59 Prime numbers, 91, 184, 218–219 Primitive recursion, 221–222, 227 Principia Mathematica (Whitehead and Russell), 152 Proof, nature of, 109 See also Deductive calculus Propositional logic, 14 Proposition symbol, 14–15

Q Quantiﬁer capture, 113 Quantiﬁers, 70 bounded, 204, 210–211 elimination of, 190–192 existential, 287, 288 Quantiﬁer symbol, universal, 80 Quotient structure, 140

R Rabin, Michael, 201 Ramiﬁed analytical sets, 306 Range (of relation), 4

Reasonable language, 142–144 See also Recursively numbered language Recursion, 32, 38–44 monotone, 224 primitive, 221–222, 227 Recursion theorem, 39–40, 41–42 Recursive functions, 247–250 normal form, 248–250 partial, 250–258, 262 reduction of decision problems, 258–260 register machines, 261–263 Recursively axiomatizable theories, 233, 240 Recursively enumerable (r.e.) relations, 233, 238–241 Recursively inseparable sets, 245 Recursively numbered language, 225 Recursive relations, 207–210, 232 See also Recursively enumerable relations Reductio ad absurdum, 119, 121 Reducts of number theory, 182–183, 193–202 Reﬂexive relations, 5 Register machines, 208, 261–263 Relation comprehension formulas, 284 Relations, 4–6 Relation universe, 301 Relay circuits, 57 Representable functions, 212–217 Representable relations, 205–206 and numeralwise determined formulas, 206, 210–212 weakly, 241–242 Re-replacement lemma, 130 Resolution, 53 Restriction, 5, 221 Rice’s theorem, 260 Rigid structure, 98

Robinson, Abraham, 173 Root of tree, 7 Rule EI, 124–125, 145 Rules of inference, 110 Rule T, 118

S Satisfaction of formulas, 23, 83–86 Satisﬁable sets, 59–60, 134 Schema, 286 Schr¨oder–Bernstein theorem, 9 Second-order logic absolute, 303 and many-sorted logic, 295–299 general structures, 299–306 language of, 282–286 Skolem functions, 145, 287–290 Segments of sequences, 4 initial, 4 terminal, 105–106 Self-reference, 234–235, 315 approach to incompleteness, 184–186 Semantics and syntax, 125 Semidecidable set, 63 Semidecision procedure, 63 Sentences, 77, 79 Sentence symbols, 14, 115 Sentential connectives, 14, 45–54 binary, 51 ternary, 51–52 unary, 51 0-ary, 50 Sentential logic compactness, 24, 59–60 connectives, 45–52 language of, 13–19 parsing algorithm, 29–33 tautologies, 23 truth assignments, 20–27 Sequence encoding and decoding, 220, 277, 281 Sequence number, 221 Sequences, ﬁnite, 4

316 Sets concept, 1–2 countable, 6 disjoint, 3 empty versus nonempty, 2 ﬁnite, 6 intersection of, 3 ordered, 93 pairwise disjoint, 3 power, 2–3 union of, 3 Set theory (ST) 152, 157, 161–162, 240–241, 270–275 G¨odel incompleteness theorems for, 274–275 language of, 70, 71 Sheffer stroke, 51 Shepherdson, John C., 261 Shepherdson–Sturgis machines, 261–263 Simpliﬁcation of formulas, 59 Single-valued relations, 5 Skolem, Thoralf, 145, 151, 293 functions, 145, 287–290 L¨owenheim–Skolem theorem, 151–155 normal form, 288–289 paradox, 152 S-m-n theorem. See Parameter theorem Soundness theorem, 66, 131 Spectrum, 101, 150, 285 Standard part, 178 Steinitz’s theorem, 158–159 Strategy for deductions, 120–126 String, 4 Strong undecidability, 237, 272–273 Structures, 80–81 cardinality of, 153–154 deﬁnability in, 90–92 deﬁnability of a class of, 92–94 general, 299–306 Sturgis, H. E., 261 Subsets, 2 Substitutability, 113 Substitution, 28, 112–114 and alphabetic variants, 126

Index lemma, 133–134 of terms, 112, 129 representability of, 228 Substructures, 95–96, 294 Sufﬁciently strong theory, 246, 267 Switching circuits, 54–59 Symbols logical, 14, 69–70 nonlogical, 14 parameter, 71 sentential connective, 14, 45–54 Symmetric relations, 5 Syntactical translation, 169–172 Syntax, arithmetization of, 224–234 Syntax and semantics, 125

Trakhtenbrot’s theorem, 151 Transitive relations, 5 Trees, 7 of deduction, 116–117 of well-formed formulas, 17, 22, 75 Trichotomy, 6, 93, 159, 194 Truth, 20 undeﬁnability of, 236, 240 Truth and models, in ﬁrst-order logic, 80–99 Truth assignments, 20–27 Truth tables, 24–26 Truth values, 20 Turing, Alan, 208, 261 Turing machine, 208 Two-valued logic, 20 Tychonoff’s theorem, 24

U T Tarski, Alfred, 152, 154, 159, 286 undeﬁnability theorem, 236, 240 Tautological equivalence, 24 Tautological implication, 23 Tautologies, 23 in ﬁrst-order languages, 114–116 representability of, 230–231 selected list of, 26–27 Term-building operations, 74 Terminal segment, 105–106 Terms, 74, 79 parsing, 106–107 representing, 226–227 unique readability of, 107 Ternary connectives, 51–52 Theorem, concept of, 110–111, 117 Theories, 155–160 axiomatizable, 156 ﬁnitely axiomatizable, 156 interpretations between, 164–172 models of, 147–162 of structures, 148, 152, 155 Total function, 250 T -predicate, 249

Ultraproducts, 142 Unary connectives, 51 Uncountable languages, 141, 153–154 Undecidability incompleteness and, 234–235 of number theory, 182–187 of set theory, 272 strong, 237, 272–273 Undeﬁnability theorem, Tarski, 236, 240 Union, 3 Unique existential quantiﬁer, 102, 165 Unique readability theorem for formulas, 108 for terms, 107 in sentential logic, 40–41 Universal formulas (∀1 ), 102 Universal quantiﬁer symbol, 68, 70, 80 Universe, of structures, 81 Unsolvability of halting problem, 254

V Valid formula, 88–89 in second-order logic, 286

Index Variables, 70, 79 bound, 80 free, 76–77 function, 282 individual, 282–283 predicate, 282 Vaught’s test. See Ło´s–Vaught test Vector spaces, 92, 99

317 W

Z

Weakly representability, 241–242 Well-deﬁned function, 165 Well-formed formulas (wffs), 12, 17–18, 75 Well-ordering, 283

Z-chains, 189–190, 197 Zermelo–Fraenkel set theory, 157, 270 0-ary connectives, 50 0-place function symbols, 70 Zorn’s lemma, 7, 60, 141