Philosophical Communications, Web Series, No 36 Dept. of Philosophy, Göteborg University, Sweden ISSN 1652-0459 Per Lindström FIRST-ORDER LOGIC May 15, 2006 PREFACE Predicate logic was created by Gottlob Frege (Frege (1879)) and first-order (predicate) logic was first singled out by Charles Sanders Peirce and Ernst Schröder in the late 1800s (cf. van Heijenoort (1967)), and, following their lead, by Leopold Löwenheim (Löwenheim (1915)) and Thoralf Skolem (Skolem (1920), (1922)). Both these contributions were of decisive importance in the development of modern logic. In the case of Frege's achievement this is obvious. However, Frege's (second-order) logic is far too complicated to lend itself to the type of (mathematical) investigation that completely dominates modern logic. It turned out, however, that the first-order fragment of predicate logic, in which you can quantify over “individuals” but not, as in Frege's logic, over sets of “individuals”, relations between “individuals”, etc., is a logic that is strong enough for (most) mathematical purposes and still simple enough to admit of general, nontrivial theorems, the Löwenheim theorem, later sharpened and extended by Skolem, being the first example. In stating his theorem, Löwenheim made use of the idea, introduced by Peirce and Schröder, of satisfiability in a domain D, i.e., an arbitrary set of “individuals” whose nature need not be specified; the cardinality of D is all that matters. This concept, a forerunner of the present-day notion of truth in a model, was quite foreign to the Frege-Peano-Russell tradition dominating logic at the time and its introduction and the first really significant theorem, the Löwenheim (-Skolem) theorem, may be said to mark the beginning of modern logic. First-order logic turned out to be a very rich and fruitful subject. The most important results, which are at the same time among the most important results of logic as a whole, were obtained in the 1920's and 30's: the Löwenheim-Skolem- Tarski theorem, the first completeness theorems (Skolem (1922), (1929), Gödel (1930)), the compactness theorem (Gödel (1930) (denumerable case), Maltsev (1936)), and the undecidability of first-order logic (Church (1936b), Turing (1936)). This period also saw the beginnings of proof theory (Gentzen (1934-35), Herbrand (1930)). In fact, the main areas of research in modern logic, model theory, computability (recursion) theory, and proof theory were all inspired by and grew out of the study of first-order logic. During most of the 1940's the subject lay fallow; logic in the 1940's was dominated by computability theory and decision problems. This lasted until the rediscovery by Henkin of the compactness theorem (Henkin (1949)) – Maltsev's work was not known in the West at the time – and the subsequent numerous contributions of Alfred Tarski, Abraham Robinson, and others in the 1950's. And since then (the theory of) first-order logic has developed into a vast and technically advanced field. But in spite of its central role in logic there still seems to be no exposition centering on first-order logic; in fact, none that covers even the material presented here. The present little book is intended to, at least partially, fill this gap in the literature. Most of the results presented in this book were obtained before 1960 and all of them before 1970. I have confined myself (in Chapter 3) to results that will (hopefully) appear meaningful and interesting even to nonlogicians. However, sometimes the proof of a result, even the fact that it can be proved, may be more interesting than the result itself. The reader I have in mind is thoroughly at home with the elementary aspects of first-order logic and, perhaps somewhat vaguely, aware of the basic concepts and results and would like to see exact definitions and full proofs of these. The reader is also assumed to be familiar with elementary set theory including simple cardinal arithmetic. Zorn's Lemma is used twice (and formulated explicitly) and definition by transfinite induction is used three times (and once in Appendix 5); that's all. In Chapter 1, §7 there are several examples of first-order theories, some of them taken from “modern algebra”. These are used in Chapter 3 to illustrate some of the model-theoretic results proved in that chapter. However, no knowledge of algebra is presupposed; the algebraic results, elementary and not so elementary, needed in these applications are stated without proof. P. L. CONTENTS Chapter 0. Introduction 1 Chapter 1. The elements of first-order logic 5 §1 Syntax of L1 5 §2 Semantics of L1 6 §3 Prenex and negation form 10 §4 Elimination of function symbols 10 §5 Skolem functions 11 §6 Logic and set theory 12 §7 Some first-order theories 13 Notes 16 Chapter 2. Completeness 17 §1 A Frege-Hilbert-type system 17 §2 Soundness and completeness of FH 20 §3 A Gentzen-type sequent calculus 26 §4 Two applications 29 §5 Soundness and completeness of GS 34 §6 Natural deduction 38 §7 Soundness and completeness of ND 42 §8 The Skolem-Herbrand Theorem 43 §9 Validity and provability 44 Notes 45 Chapter 3. Model theory 46 §1 Basic concepts 46 §2 Compactness and cardinality theorems 49 §3 Elementary and projective classes 52 §4 Preservation theorems 54 §5 Interpolation and definability 58 §6 Completeness and model completeness 61 §7 The Fraïssé-Ehrenfeucht criterion 66 §8 Omitting types and ℵ0-categoricity 69 §9 Ultraproducts 75 §10 Löwenheim-Skolem theorems for two cardinals 80 §11 Indiscernibles 83 §12 An illustration 87 §13 Examples 87 Notes 93 Chapter 4. Undecidability 96 §1 An unsolvable problem 96 §2 Undecidability of L1 99 §3 The Incompleteness Theorem 103 §4 Completeness and decidability 105 §5 The Church-Turing thesis 106 Notes 106 Chapter 5. Characterizations of first-order logic 108 §1 Extensions of L1 108 §2 Properties of logics 110 §3 Characterizations 112 Notes 118 Appendix 1 119 Appendix 2 125 Appendix 3 128 Appendix 4 131 Appendix 5 133 Appendix 6 137 Appendix 7 139 References 140 Index 144 Symbols 148 1 0. INTRODUCTION Suppose you are interested in a certain mathematical object (structure, model) M, say, the sequence of natural numbers 0, 1, 2, ... or the Euclidean plane or the family of all sets. You want to know, or be able to find out, for its own sake or for the sake of application, what is true and what is not about M. The first thing you have to do is then to decide on certain basic (primitive) concepts in terms of which you are going to formulate statements about M. In the case of the natural numbers the natural choice is addition and multiplication. In the case of the Euclidean plane the concepts point, (straight) line, and lies on (a relation between points and lines) are natural choices and there are others. Finally, in set theory the obvious choice, at least since Cantor, is the element relation. The primitive concepts are not defined (in terms of other concepts) – you can't define everything – but should be sufficiently clear, possibly on the basis of informal explanation. Additional concepts such as exponentiation and prime number or triangle and parallel with or function and ordinal number can then be introduced by definition. Your goal is to be able to prove nontrivial theorems about M. Since you cannot prove everything, you have to start by accepting certain statements about M as true without proof. These are your axioms. The idea, which goes back to Euclid, is then to prove theorems by showing that they follow from, or are implied by, the axioms. But “follow from” in what sense? It is in an attempt to answer this question, and related questions, that logic enters the stage. In (mathematical) logic we want to be able to investigate, by mathematical means, mathematical statements, theories, and families of theories in much the same way as numbers are studied in number theory, points, straight lines, circles, etc. in geometry, sets in set theory, topological spaces in topology, etc. But mathematical statements and theories in their usual form and the relation follows from are not sufficiently well-defined (or explicit) to be amenable to investigation of this nature. Thus, the first thing we shall have to do is replace such statements and theories by other entities sufficiently well-defined to form the subject matter of mathematical theorizing. This is achieved by formalization. To formalize a theory T you first introduce a formal (artificial) language, or skeleton of a language, l with a perfectly precise (and perspicuous) syntax; in other words, the “alphabet” (set of primitive symbols) including the mathematical symbols, for example, + and . in the case of number theory and ∈ in the case of set theory, should be explicitly given and the rules of formation, i.e., 2 definitions of “term” (noun phrase), “formula”, “sentence” (formula without free variables), etc. of l should be explicitly stated (cf. Chapter 1, §1). In formulas we need, in addition to the mathematical symbols, among other things, certain logical symbols such as ¬ (not), ∧ (and), ∃ (there exists), = (equal to). The next step is then to define a suitable semantical interpretation of l. Thus, for any sentence ϕ of l and any model M (appropriate) for l, it should be explicitly defined what it means to say that ϕ is true in M, or M is a model of ϕ (cf. Chapter 1, §2). In this definition the meaning of the logical symbols is constant (independent of M) whereas the meaning of the mathematical symbols is determined by M. We are going to need true in for all models and not just for the model we are particularly interested in.