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    Model Theory: Third Edition
    Model Theory: Third Edition
    Model Theory: Third Edition
    Ebook994 pages8 hours

    Model Theory: Third Edition

    By C.C. Chang and H. Jerome Keisler

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    Model theory deals with a branch of mathematical logic showing connections between a formal language and its interpretations or models. This is the first and most successful textbook in logical model theory. Extensively updated and corrected in 1990 to accommodate developments in model theoretic methods — including classification theory and nonstandard analysis — the third edition added entirely new sections, exercises, and references.
    Each chapter introduces an individual method and discusses specific applications. Basic methods of constructing models include constants, elementary chains, Skolem functions, indiscernibles, ultraproducts, and special models. The final chapters present more advanced topics that feature a combination of several methods. This classic treatment covers most aspects of first-order model theory and many of its applications to algebra and set theory.
    LanguageEnglish
    PublisherDover Publications
    Release dateOct 3, 2013
    ISBN9780486310954
    Model Theory: Third Edition

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      Model Theory - C.C. Chang

      symbols

      CHAPTER 1

      INTRODUCTION

      1.1 What is model theory?

      Model theory is the branch of mathematical logic which deals with the relation between a formal language and its interpretations, or models. We shall concentrate on the model theory of first-order predicate logic, which may be called ‘classical model theory’.

      Let us now take a short introductory tour of model theory. We begin with the models which are structures of the kind which arise in mathematics. For example, the cyclic group of order 5, the field of rational numbers, and the partially-ordered structure consisting of all sets of integers ordered by inclusion, are models of the kind we consider. At this point we could, if we wish, study our models at once without bringing the formal language into the picture. We then would be in the area known as universal algebra, which deals with homomorphisms, substructures, free structures, direct products, and the like. The line between universal algebra and model theory is sometimes fuzzy; our own usage is explained by the equation

      To arrive at model theory, we set up our formal language, the first-order logic with identity. We specify a list of symbols and then give precise rules by which sentences can be built up from the symbols. The reason for setting up a formal language is that we wish to use the sentences to say things about the models. This is accomplished by giving a basic truth definition, which specifies for each pair consisting of a sentence and a model one of the truth values true or false. The truth definition is the bridge connecting the formal language with its interpretation by means of models. If the truth value ‘true’ goes with the sentence φ , we say that φ is true in is a model of φ. Otherwise we say that φ is false in is not a model of φis a model of a set Σ is a model of each sentence in the set Σ.

      What kinds of theorems are proved in model theory? We can already give a few examples. Perhaps the earliest theorem in model theory is Löwenheim’s theorem (Löwenheim, 1915): If a sentence has an infinite model, then it has a countable model. Another classical result is the compactness theorem, due to Gödel (1930) and Malcev (1936): if each finite subset of a set Σ of sentences has a model, then the whole set Σ has a model. As a third example, we may state a more recent result, due to Morley (1965). Let us say that a set Σ of sentences is categorical in power α iff there is, up to isomorphism, exactly one model of Σ of power α. Morley’s theorem states that, if Σ is categorical in one uncountable power, then Σ is categorical in every uncountable power.

      These theorems are typical results of model theory. They say something negative about the ‘power of expression’ of first-order predicate logic. Thus Löwenheim’s theorem shows that no consistent sentence can imply that a model is uncountable. Morley’s theorem shows that first-order predicate logic cannot, as far as categoricity is concerned, tell the difference between one uncountable power and another. And the compactness theorem has been used to show that many interesting properties of models cannot be expressed by a set of first-order sentences – for instance, there is no set of sentences whose models are precisely all the finite models.

      The three theorems we have stated also say something positive about the existence of models having certain properties. Indeed, in almost all of the deeper theorems in model theory the key to the proof is to construct the right kind of a model. For instance, look again at Löwenheim’s theorem. To prove that theorem, we must begin with an uncountable model of a given sentence and construct from it a countable model of the sentence. Likewise, to prove the compactness theorem we must construct a single model in which each sentence of Σ is true. Even Morley’s theorem depends vitally on the construction of a model. To prove it we begin with the assumption that Σ has two different models of one uncountable power and construct two different models of every other uncountable power.

      There are a small number of extremely important ways in which models have been constructed. For example, for various purposes they can be constructed from individual constants, from functions, from Skolem terms, or from unions of chains. These constructions give the subject of model theory unity. To a large extent, we have organized this book according to these ways of constructing models.

      Another point which gives model theory unity is the distinction between syntax and semantics. Syntax refers to the purely formal structure of the language – for instance, the length of a sentence and the collection of symbols occurring in a sentence, are syntactical properties. Semantics refers to the interpretation, or meaning, of the formal language – the truth or falsity of a sentence in a model is a semantical property. As we shall soon see, much of model theory deals with the interplay of syntactical and semantical ideas.

      We now turn to a brief historical sketch. The mathematical world was forced to observe that a theory may have more than one model in the 19th century, when Bolyai and Lobachevsky developed non-Euclidean geometry, and Riemann constructed a model in which the parallel postulate was false but all the other axioms were true. Later in the 19th century, Frege formally developed the predicate logic, and Cantor developed the intuitive set theory in which our models live.

      Model theory is a young subject. It was not clearly visible as a separate area of research in mathematics until the early 1950’s. However, its historical roots go back to the older subjects of logic, universal algebra, and set theory – and some of the early work, such as Löwenheim’s theorem, is now classified as model theory. Other important early developments which contributed to the theory are: the extension of Löwenheim’s theorem by Skolem (1920) and Tarski; the completeness theorem of Gödel (1930) and its generalization by Malcev (1936); the characterization of definable sets of real numbers, the rigorous definition of the truth of a sentence in a model, and the study of relational systems by Tarski (1931, 1933, 1935a); the construction of a nonstandard model of number theory by Skolem (1934); and the study of equational classes initiated by Birkhoff (1935). Model theory owes a great deal to general methods which were originally developed for special purposes in older branches of mathematics. We shall come across many instances of this in our book; to mention just one, the important notion of a saturated model (Chapter 5) goes back to the ηα-structures in the theory of simple order, due to Hausdorff (1914). The subject grew rapidly after 1950, stimulated by the papers of Henkin (1949), Tarski (1950), and Robinson (1950). The phrase ‘theory of models’ is due to Tarski (1954). Today the literature in the subject is quite extensive. There is a rather complete bibliography in Addison, Henkin and Tarski (1965). In recent years, the theory of models has been applied to obtain significant results in other fields, notably set theory, algebra and analysis. However, until now only a tiny part of the potential strength of model theory has been used in such applications. It will be interesting to see what happens when (and if) the full strength is used.

      1.2 Model theory for sentential logic

      In our introduction, Section 1.1, we gave a general idea of the flavor of model theory, but we were not yet ready to give many details. We shall now come down to earth and give a rigorous treatment of model theory for a very simple formal language, sentential logic (also known as propositional calculus). We shall quickly develop this ‘toy’ model theory along lines parallel to the much deeper model theory for predicate logic. The basic ideas are the decision procedure via truth tables, due to Post (1921), and Lindenbaum’s theorem with the compactness theorem which follows. This section will give a preview of what lies ahead in our book.

      We are assuming (see Preface) that the reader is already thoroughly familiar with sentential, and even predicate, logic. Thus we shall feel free to proceed at a fairly rapid pace. Nevertheless, we shall start from scratch, in order to show what sentential logic looks like when it is developed in the spirit of model theory.

      of simple statements, and the compound statements built up from them. At the most intuitive level, an intended interpretation of these statements is a ‘possible world’, in which each statement is either true or false. We wish to replace these intuitive interpretations by a collection of precise mathematical objects which we may use as our models. The first thing which comes to mind is a function F which associates with each simple statement S one of the truth values ‘true’ or ‘false’. Stripping away the inessentials, we shall instead take a model to be a subset A ; the idea is that S A indicates that the simple statement S is true, and S A indicates that the simple statement S is false.

      1.2.1. By a model A for we simply mean a subset A .

      itself.

      We now set up the sentential logic as a formal language. The symbols of our language are as follows:

      (not);

      parentheses ), (;

      of sentence symbols.

      stand for the words used to combine simple statements into compound statements. Formally, the sentences are defined as follows:

      1.2.2.

      (i). Every sentence symbol S is a sentence.

      (ii). If φ φ) is a sentence.

      (iii). If φ, ψ are sentences, then (φ ψ) is a sentence.

      (iv). A finite sequence of symbols is a sentence only if it can be shown to be a sentence by a finite number of applications of (i)-(iii).

      sentence of may be restated as a recursive definition based on the length of a finite sequence of symbols:

      A single symbol is a sentence iff it is a sentential symbol; a sequence φ of symbols of length n > 1 is a sentence iff there are sentences ψ and θ of length less than n such that φ ψ) or (ψ θ).

      Alternatively, our definition may be restated in set-theoretical terms:

      is the least set Σ such that each sentence symbol S belongs to Σ and, whenever ψ, θ are in Σ, ψ), (ψ θ) belong to Σ.

      No matter how we may think of sentences, the important thing is that properties of sentences can only be established through an induction based on 1.2.2. More precisely, to show that every sentence φ has a given property P, we must establish three things: (1) Every sentence symbol S has the property P; (2) if φ ψ) and ψ has the property P, then φ has the property P; (3) if φ is (ψ θ) and ψ, θ have the property P, then φ has the property P. (The reader may check his understanding of this point by proving through induction that every sentence φ has the same number of right parentheses as it has left parentheses.)

      are there? This depends on the number of sentence symbols S . Of course, not every finite sequence of symbols is a sentence; for instance, (SS)S3 and SSof sentence symbols has uncountable cardinal αalso has power α.

      Let us pause briefly to explain the role of the Greek letters φ, ψ, Σ, etc. In the above paragraphs we have used the lower case Greek letters (φ, ψ, θ. These letters were needed in order to write down the definition of a sentence. From now on, we shall be much more interested in sentences than in arbitrary finite sequences of symbols. We shall hereafter use the lower case Greek letters φ, ψ, θ. The situation is similar to elementary arithmetic, where we study natural numbers 0, 1,2,3, …, but much of the time we write down letters like m, n, x, y, … as names for arbitrary natural numbers. Just as in arithmetic where we write things like m = x + y, we shall now write, for example, φ = (ψ θ) to express the fact that φ and (ψ θ) are the same sentence. In the above paragraphs we also used capital Greek letters Σ, Γ. The symbols (φ, ψ, θ …, Σ, Γ, … are not .

      (or), → (implies), and ↔ (if and only if) are abbreviations defined as follows:

      , → and ↔ could just as well have been included in our list of symbols as three more connectives. However, there are certain advantages to keeping our list of symbols short. For instance, 1.2.2 and proofs by induction based on it are shorter this way. At the other extreme, we could have managed with only a single connective, whose English translation is ‘neither … nor …’. We did not do this because ‘neither … nor …’ is a rather unnatural connective.

      S Sφ ψ → θ φ φψ) → (θ φ).

      to denote both the set of sentence symbols and the language built on these symbols. There is no fear of confusion in this double usage since the language is determined uniquely, modulo the connectives, by the sentence symbols.

      and its models, with the definition of the truth of a sentence in a model. We shall express the fact that a sentence φ is true in a model A succinctly by the special notation

      The relation A φ is defined as follows:

      1.2.3.

      (i). If φ is a sentence symbol S, then A φ holds if and only if S A.

      (ii). If φ is ψ θ, then A φ if and only if both A ψ and A θ.

      (iii). If φ ψ, then A φ iff it is not the case that A ψ.

      When A φ, we say that φ is true in A, or that φ holds in A, or that A is a model of φ. When it is not the case that A φ, we say that φ is false in A, or that φ fails in A. The above definition of the relation A φ is an example of a recursive definition based on 1.2.2. The proof that the definition is unambiguous for each sentence φ is, of course, a proof by induction based on 1.2.2.

      An especially important kind of sentence is a valid sentence. A sentence φ is called validφ, iff φ , that is, iff A φ for all A. Some notions closely related to validity are mentioned in the exercises.

      infinite we have to examine uncountably many different infinite models A in order to find out whether a sentence φ is valid. This is because validity is a semantical notion, defined in terms of models. However, as the reader surely knows, there is a simple and uniform test by which we can find out in only finitely many steps whether or not a given sentence φ is valid.

      This decision procedure for validity is based on a syntactical notion, the notion of a tautology. Let φ be a sentence such that all the sentence symbols which occur in φ are among the n + 1 symbols S0, S1, …, Sn. Let a0, a1, …, an be a sequence made up of the two letters t, f. We shall call such a sequence an assignment.

      1.2.4. The value of a sentence φ for the assignment a0, …, an is defined recursively as follows:

      (i). If φ is the sentence symbol Sm, m ≤ n, then the value of φ is am.

      (ii). If φ ψ, then the value of φ is the opposite of the value of ψ.

      (iii). If φ is ψ θ, then the value of φ is t if the values of ψ and θ are both t, and otherwise the value of φ is f.

      Note how similar Definitions 1.2.3 and 1.2.4 are. The only essential difference is that 1.2.3 involves an infinite model A, while 1.2.4 involves only a finite assignment a0, …, an.

      1.2.5. Let φ be a sentence and let S0, …, Sn be all the sentence symbols occurring in φ. φ is said to be a tautologyφ, iff φ has the value t for every assignment a0, …, an.

      is used for syntactical ideas.

      The value of a sentence φ for an assignment a0, …, an may be very easily computed. We first find the values of the sentence symbols occurring in φ and then work our way through the smaller sentences used in building up the sentence φ. A table showing the value of φ for each possible assignment a0, …, an is called a truth table of φ. We shall assume that truth tables are already quite familiar to the reader, and that he knows how to construct a truth table of a sentence. Truth tables provide a simple and purely mechanical procedure to determine whether a sentence φ is a tautology – simply write down the truth table for φ and check to see whether φ has the value t for every assignment.

      PROPOSITION 1.2.6. Suppose that all the sentence symbols occurring in φ are among S0, S1, …, Sn. Then the value of φ for an assignment a0, a1, an, …, an+m is the same as the value of φ for the assignment a0, a1, …, an.

      We now prove the first of a series of theorems which state that a certain syntactical condition is equivalent to a semantical condition.

      φ if and only if φ; in words, a sentence is a tautology if and only if it is valid.

      PROOF. Let φ be a sentence and let all the sentence symbols in φ be among S0, …, Sn. Consider an arbitrary model A. For m = 0, 1, n, put am = t if Sm A, and am = f if Sm A. This gives us an assignment a0, a1, …, an. We claim:

      (1) A φ if and only if the value of φ for the assignment a0, a1, …, an is t. This can be readily proved by induction. It is immediate if φ is a sentence symbol Sm. Assuming that (1) holds for φ = ψ and for φ = θ, we see at once that (1) holds for φ ψ and φ = ψ θ.

      Now let S0, …, Sn be all the sentence symbols occurring in φ. If φ is a tautology, then by (1), φ is valid. Since every assignment a0, a1, …, an can be obtained from some model A, it follows from (1) that, if φ is valid, then φ

      φ now can be used to decide whether φ is valid, because the proof is always the same – we simply look at the truth table.

      .

      The Rule of Detachment (or Modus Ponens) states:

      We say that φ is inferred from ψ, θ by detachment iff θ is the sentence ψ → φ.

      Now consider a finite or infinite set Σ .

      A sentence φ is deducible from Σ, in symbols Σ φ, iff there is a finite sequence ψ0, ψ1, …, ψn sentences such that φ = ψn and each sentence ψm is either a tautology, belongs to Σ, or is inferred from two earlier sentences of the sequence by detachment. The sequence ψ0, ψ1, …, ψn called a deduction of φ from Σ. Note that φ is deducible from the empty set of sentences if and only if φ is a tautology.

      We shall say that Σ is inconsistent iff we have Σ φ for all sentences φ. Otherwise, we say that Σ is consistent. Finally, we say that Σ is maximal consistent iff Σ is consistent, but the only consistent set of sentences which includes Σ is Σ itself. The proposition below contains facts which can be found in most elementary logic texts.

      PROPOSITION 1.2.8.

      (i). If Σ is consistent and Γ is the set of all sentences deducible from Σ, then Γ is consistent.

      (ii). If Σ is maximal consistent and Σ φ, then φ Σ.

      (iii). Σ is inconsistent if and only if Σ S S (for any S ).

      (iv). (Deduction Theorem). If , then Σ ψ → φ.

      LEMMA 1.2.9 (Lindenbaum’s Theorem). Any consistent set Σ of sentences can be enlarged to a maximal consistent set Γ of sentences.

      in a list, φ0, φ1, φ2, …, φα, …. The order in which we list them is immaterial, as long as the list associates in a one–one fashion an ordinal number with each sentence. We shall form an increasing chain

      . Otherwise define Σ1 = Σ. At the αis consistent, and otherwise define Σα+1 = Σα. Now let Γ be the union of all the sets Σα.

      We claim that Γ is consistent. Suppose not. Then there is a deduction ψ0, ψ1, …, ψp of sentence S S from Γ (see Proposition 1.2.8). Let θ1, …, θq be all the sentences in Γ which are used in this deduction. We may choose α so that all of θ1, …, θq belong to Σα. But this means that Σα is inconsistent (again see Proposition 1.2.8), which is a contradiction.

      Having shown that Γ is consistent, we next claim that Γ is maximal consistent. For suppose Δ , and hence Δ = Γ

      LEMMA 1.2.10. Suppose Γ is a maximal consistent set of sentences in . Then:

      (i). For each sentence φ, exactly one of the sentences φφ belongs to Γ.

      (ii). For each pair of sentences φ, ψ, φ ψ belongs to Γ if and only if both φ and ψ belong to Γ.

      We leave the proof as an exercise.

      Now consider a set Σ . We shall say that A is a model of Σ, A Σ, iff every sentence φ Σ is true in A. Σ is said to be satisfiable iff it has at least one model. We now prove the most important theorem of sentential logic, which is a criterion for a set Σ to be satisfiable.

      THEOREM 1.2.11 (Extended Completeness Theorem). A set Σ of sentences of is consistent if and only if Σ is satisfiable.

      PROOF. Assume first that Σ is satisfiable, and let A Σ. We show that every sentence deducible from Σ holds in A. Let ψ0, ψ1, …, ψn be a deduction of ψn from Σ. Let m ≤ n. If ψm Σ or if ψm is a tautology, then ψm holds in A. If ψm is inferred from two sentences ψp, ψp → ψm which hold in A, then ψm must clearly hold in A. It follows by induction on m that each of the sentences ψ0, ψ1, …, ψn holds in A. Since S S does not hold in A, it is not deducible from Σ, so Σ is consistent.

      Now assume that Σ is consistent. By Lindenbaum’s theorem we enlarge Σ to a maximal consistent set Γ.

      We now construct a model of Σ. Let A be the set of all sentence symbols S such that S Γ. We show by induction that, for each sentence φ,

      By definition, (1) holds when φ is a sentence symbol Sn. Lemma 1.2.10(i) guarantees that, if (1) holds when φ = ψ, then (1) holds when φ ψ. Lemma 1.2.10(ii) guarantees that, if (1) holds when φ = ψ and when φ = θ, then (1) holds when φ = ψ θ. From (1) it follows that A Γ

      We can obtain a purely semantical corollary. Σ is said to be finitely satisfiable iff every finite subset of Σ is satisfiable.

      COROLLARY 1.2.12 (Compactness Theorem). If Σ is finitely satisfiable, then Σ is satisfiable.

      PROOF. Suppose Σ is not satisfiable. Then by the extended completeness theorem Σ is inconsistent. Hence, Σ S S. In the deduction of the sentence S S from Σ only a finite set Σ0 of sentences of Σ is used. It follows that ΣS S, so Σ0 is inconsistent. Then Σ0 is not satisfiable, so Σ

      Note that the converse of the compactness theorem is trivially true, i.e., every satisfiable set of sentences is finitely satisfiable.

      We say that φ is a consequence of Σ, in symbols Σ φ, iff every model of Σ is a model of φ. The reader is asked to prove Exercises 1.2.3–1.2.6 as well as the following:

      COROLLARY 1.2.13

      (i). Σ φ if and only if Σ φ.

      (ii). If Σ φ, then there is a finite subset Σ0 of Σ such that Σφ.

      We shall conclude our model theory for sentential logic with a few applications of the compactness theorem. In these applications, the true spirit of model theory will appear, but at a very rudimentary level. Since we shall often wish to combine a finite set of sentences into a single sentence, we shall use expressions like

      and

      In these expressions the parentheses are assumed, for the sake of definiteness, to be associated to the right; for instance,

      First we introduce a bit more terminology. A set Γ of sentences is called a theory. A theory is said to be closed iff every consequence of Γ belongs to Γ. A set Δ of sentences is said to be a set of axioms for a theory Γ iff Γ and Δ have the same consequences. A theory is called finitely axiomatizable of all consequences of Γ is the unique closed theory which has Γ as a set of axioms.

      PROPOSITION 1.2.14. Δ is a set of axioms for a theory Γ if and only if Δ has exactly the same models as Γ.

      COROLLARY 1.2.15. Let Γ1 and Γ2 be two theories such that the set of all models of Γ2 is the complement of the set of all models of Γ1. Then Γ1 and Γ2 are both finitely axiomatizable.

      PROOF. The set Γ1 ∪ Γsuch that Δ1 ∪ Δ2 is not satisfiable. If A Δ1, then A is not a model of Γ2, and consequently A Γ1. It follows by Proposition 1.2.14 that Δ1 is a finite set of axioms for Γ1. Similarly Δ2 is a finite set of axioms for Γ

      The next group of theorems shows connections between mathematical operations on models and syntactical properties of sentences. The first result of this group concerns positive sentences. A sentence φ is said to be positive iff φ . For example, (S(SSSS4 and S3 ↔ S3 are not positive. A set Σ of sentences is called increasing iff A Σ and A ⊂ B implies B Σ.

      THEOREM 1.2.16.

      (i). A ⊂ B if and only if every positive sentence which holds in A holds in B.

      (ii). A consistent theory Γ is increasing if and only if Γ has a set ofpositive axioms.

      (iii). A sentence φ is increasing if and only if either φ is equivalent to a positive sentence, φ is valid, or φ is valid.

      PROOF. (i). The fact that, if A ⊂ B, then every positive sentence which holds in A holds in B, is proved by induction. First, every sentence symbol which holds in A holds in B, because of 1.2.3(i) and A ⊂ B. Using 1.2.3(ii) and Exercise 1.2.2, it can be checked that, if the condition ‘if φ holds in A, then φ holds in B’ is true when φ = ψ, and when φ = θ, then it is also true when φ = ψ θ and when φ = ψ θ. Hence that condition is true for every positive sentence φ.

      Suppose that every positive sentence which holds in A holds in B. In particular, for each S , if A S, then B S. Thus, if S A, then S B, so A ⊂ B. This proves (i).

      (ii). Now let Γ be a consistent increasing theory. Let Δ be the set of all positive consequences of Γ. Suppose B Δ. Let Σ φ such that φ is positive and B φφφn Σ. Then the sentence φφn is a positive sentence which fails in B. Hence φφn does not belong to Δ and is not a consequence of Γ. Thus the set Γ φφn} is satisfiable, and the set Γ ∪ Σ is finitely satisfiable. By the compactness theorem, Γ ∪ Σ has a model, say A. Now for every positive sentence φ which fails in Bφ Σ, so φ fails in A. Thus every positive sentence holding in A holds in B, and by (i), A ⊂ B. Since A Γ and Γ is increasing, we have B Γ. We conclude that every model of Δ is a model of Γ. But Δ , and therefore Δ is a set of positive axioms for Γ.

      Conversely, if Γ has a set of positive axioms, then it follows from (i) that Γ is increasing.

      (iii). Let φ be an increasing sentence. We may assume further that φ is satisfiable. If Γ is the set of all consequences of φ, then by (ii) Γ has a positive set Δ of axioms. Now φ Γ, so Δ φ, and by Corollary 1.2.13 there is a finite subset {ψ1, …, ψn} of Δ such that {ψ1, …, ψnφ. If n = 0, then φ is valid. Let n > 0. Each ψm is in Δ and thus in Γ, so each ψm is a consequence of φ. It follows that φ is equivalent to the positive sentence ψψn.

      A completely trivial fact which is analogous to part (i) of the above theorem is: A = B if and only if every sentence which holds in A holds in B. We shall see later on in this book that the situation is very different in predicate logic, where a maximal consistent theory ordinarily does not even come close to characterizing a single model. This is one thing which makes model theory for predicate logic so much more interesting and difficult than model theory for sentential logic.

      We now turn to another kind of sentence. By a conditional sentence we mean a sentence φφn, where each φi is of one of the following three kinds:

      (1) S,

      .

      A set Σ of sentences is said to be preserved under finite intersections iff A Σ and B Σ implies A ∩ B Σ. Σ is said to be preserved under arbitrary intersections iff for every nonempty set {Ai : i I} of models of Σ is also a model of Σ.

      LEMMA 1.2.17. A theory Γ is preserved under finite intersections if and only if Γ is preserved under arbitrary intersections.

      PROOF. Let Γ be preserved under finite intersections, let {Ai : i I} be a nonempty set of models of Γ. Let Σ be the set of all sentences of the form S S which hold in B. We show that Γ ∪ Σ is satisfiable. Let Σ0 be an arbitrary finite subset of Σ, and let the negative sentences in ΣSSp. If p = 0, all the sentences in Σ0 are positive, and each of the models Ai is a model of Σ0, because B ⊂ Ai. Let p from among the Aiis a model of Σ0; since Γ is preserved under finite intersections, A is also a model of Γ. We have shown that Γ ∪ Σ is finitely satisfiable. By the compactness theorem, Γ ∪ Σ has a model. But the only model of Σ is B, so B is a model of Γ

      In view of the above lemma, we may as well simply say from now on that Γ is preserved under intersections, since it makes no difference whether we say finite or arbitrary intersections.

      THEOREM 1.2.18.

      (i). A theory Γ is preserved under intersections if and only if Γ has a set of conditional axioms.

      (ii). A sentence φ is preserved under intersections if and only if φ is equivalent to a conditional sentence.

      PROOF. (i). We leave to the reader the proof that every conditional sentence (and hence every set of conditional sentences) is preserved under intersections.

      Conversely, let Γ be preserved under intersections. Consider the set Δ of all conditional consequences of Γ. It suffices to show that every model of Δ is a model of Γ. Let B be an arbitrary model of Δ. For each T − B, let ΣT be the set of all sentences of the form

      which hold in BT itself be in ΣT. We first note that the conjunction of finitely many sentences in ΣT is again equivalent to a sentence in ΣT. Consider a sentence φ ΣTφ is ciearly equivalent to a conditional sentence ψ either of the form S or of the form

      But ψ fails in B, so ψ does not belong to Δ. This means that ψφ, is not a consequence of Γ, and it follows that Γ ∪ {φ} is satisfiable. Since ΣT is, up to equivalence, closed under finite conjunction, we see that Γ ∪ ΣT is finitely satisfiable. Applying the Compactness Theorem, we may choose a model AT of Γ ∪ ΣT.

      For each T − B, we have T AT and B ⊂ AT− B is not empty, then

      Since each AT is a model of Γ and Γ is closed under intersections, we have B = Γ. In the remaining case B , we let Σ be the set of all sentences of the form

      Arguing as before, we find that Γ ∪ Σ is finitely satisfiable and thus has a model. But B is the only model of Σ, so again B is a model of Γ.

      We have now shown that every model of Δ is a model of Γ, and it follows that Δ is a set of conditional axioms for Γ.

      (ii). This follows from (i) by an argument similar to the last part of the proof of

      We conclude with a table which summarizes the semantical and syntactical notions that we have shown to be equivalent (some of these are done in the exercises).

      TABLE 1.2.1

      EXERCISES

      1.2.1. Let A be a model such that S, T A and U, V − A. Which of the following sentences are true in A?

      1.2.2. Show that, if φ = ψ θ, then A φ if and only if A ψ or A θ or both. Concoct similar rules for A ψ → θ and A ψ ↔ θ.

      1.2.3. A sentence φ is satisfiable iff it has at least one model. Show that φ φ is not valid.

      1.2.4. A sentence φ is a consequence of another sentence ψ, in symbols ψ φ, iff every model of ψ is a model of φ. Show that ψ φ ψ → φ.

      1.2.5. Two sentences φ and ψ are (semantically) equivalent iff they have exactly the same models. Show that φ and ψ φ ↔ ψ.

      1.2.6. Prove that if φ is countable, then the set of all models of φ has the cardinal number of the continuum.

      1.2.7* (Interpolation Theorem). Assume that φ ψ. Show that either (i) φ is refutable, (ii) ψ is valid, or (iii) there exists a sentence θ such that φ θ, θ ψ, and every sentence symbol which occurs in θ also occurs in both φ and ψ.

      1.2.8. Prove Proposition 1.2.6.

      1.2.9*

      (i). For every finite set K of models, there is a set Σ of sentences such that K is the set of all models of Z.

      (ii). Give an example of a set Σ of sentences such that the set of all models of Σ is countably infinite.

      (iii). Give an example of a countable set of models which cannot be represented as the set of all models of some set of sentences.

      is countable.

      1.2.10. If Σ φ for all φ Γ and if Σ ∪ Γ θ, then Σ θ.

      1.2.11. Prove that the set of all non-models of Σ .

      1.2.12. Show that no positive sentence is valid and no positive sentence is refutable.

      1.2.13. A theory Γ is said to be complete iff for every sentence φ, exactly one of Γ φ, Γ φ holds. For any set Σ of sentences, the following are equivalent:

      (i). The set of consequences of Σ is maximal consistent.

      (ii). Σ is a complete theory.

      (iii). Σ has exactly one model.

      (iv). There is a model A such that for all φ, Σ φ iff A φ.

      1.2.14. Let Γ be a consistent theory and let B . Prove that B is a model of the set of all positive consequences of Γ if and only if there is a model A of Γ such that A ⊂ B.

      1.2.15. Show that every conditional sentence is preserved under intersections.

      1.2.16. State and prove the analogue of Exercise 1.2.14 for intersections and conditional sentences.

      1.2.17*. Formulate and prove a result like Theorem 1.2.18 for unions of sets of models.

      1.2.18. A set Σ of sentences is said to be independent iff, for each σ Σ, σ is not a consequence of Σ − {σis countable, then every theory Γ has an independent set of axioms.

      [Hint: Show that Γ .]

      | = ω1 is very much easier than the general case, but still a challenge.)

      1.3 Languages, models and satisfaction

      We begin here the development of first-order languages in a way parallel to the treatment of sentential logic in . The precise formulation of this definition is much more of a challenge in first-order logic than it was for sentential logic. At the end of this section, we state the completeness and compactness theorems (Theorems 1.3.20–1.3.22), but the proofs of these theorems are deferred until the next chapter.

      We first establish a uniform notation and set of conventions for such languages and their models. A language is a collection of symbols. These symbols are separated into three groups, relation symbols, function symbols and (individual) constant symbolswill be denoted by capital Latin letters P, F, with subscripts. Lower case Latin letters cas follows:

      Each relation symbol P is assumed to be an n-placed relation for some integer n ≥ 1, depending on P. Similarly, each function symbol F is an m-placed function symbol, where m ≥ 1 and m depends on F, etc. If the symbols of the language are quite standard, as for example + for addition, ≤ for an order relation, etc., we shall simply write

      for such languages. The number of places of the various kinds of symbols is understood to follow the standard usage. The power, or cardinal ||, is defined as

      || is countable or uncountable.

      is an expansion is a reduction is said to be a simple expansion ∪ X, where X is the set of new symbols.

      in Section 1.2. There, each S has rather simple properties, as the reader discovered. This time, each n-placed relation symbol has as its intended interpretations all n-placed relations among the objects, each m-placed function symbol has as its intended interpretations all mconsists, first of all, of a universe A, a nonempty set. In this universe, each n-placed P corresponds to an n-placed relation R ⊂ An on A, each m-placed F corresponds to an m-placed function G : Am → A on A, and each constant symbol c corresponds to a constant x A. This correspondence is given by an interpretation to appropriate relations, functions and constants in A. A model for is a pair 〈A, etc., are precisely the sets B′, B″, Bi, Bj.

      Note that in a given universe A and R, R, respectively. We say that R′ is the corresponding relation to R , i.e.

      We introduce similar conventions as regards the functions and constants. When

      in displayed form as

      are familiar, we shall agree to use, for instance,

      . We may resort to

      if the context of the discussion requires it.

      ∪ X by giving appropriate interpretations for the symbols in X′ is any interpretation for the symbols of X , and X is an expansion is the reduct ∪ X . The processes of expansion and reduction do not change the universe of the model.

      The cardinal, or poweris the cardinal |Ais said to be finite, countable or uncountable if |A| is finite, countable or uncountable. Note that on a finite universe A|.

      We next introduce some simple but basic notions and operations on models. The reader should go through the exercises at the end of this section in order to be familiar with them.

      are isomorphic iff there is a 1-1 function f mapping A onto A′ satisfying:

      (i). For each n-placed relation R and the corresponding relation R,

      for all x1, …, xn in A.

      (ii). For each m-placed function G and the corresponding function G,

      for all x1, …, xm in A.

      (iii). For each constant x and the corresponding constant x,

      A function f that satisfies the above is called an isomorphism of onto , or an isomorphism between and to denote that f to denote the isomorphism relation , then |A| = |B|. Indeed, unless we wish to consider the particular structure of each element of A or Bare the same if they are isomorphic.

      is called a submodel if A′ ⊂ A and:

      (i). Each n-placed relation Ris the restriction to A′ of the corresponding relation R , i.e., R′ = R ∩ (A′)n.

      (ii). Each m-placed function Gis the restriction to A′ of the corresponding function G .

      .

      , then |A| ≤ |Bis an extension .

      is isomorphically embedded in and an isomorphism f such that f . In this case we call the function f an isomorphic embedding of in .

      , we need the following logical symbols in Section 1.2.):

      (identity).

      are called terms. They are defined as follows:

      1.3.1.

      (i). A variable is a term.

      (ii). A constant symbol is a term.

      (iii). If F is an m-placed function symbol and t1, …, tm are terms, then F(t1 … tm) is a term.

      (iv). A string of symbols is a term only if it can be shown to be a term by a finite number of applications of (i)–(iii).

      The atomic formulas are strings of the form given below:

      1.3.2.

      (i). tt2 is an atomic formula, where t1 and t.

      (ii). If P is an n-placed relation symbol and t1, …, tn are terms, then P(t1 … tn) is an atomic formula.

      Finally, the formulas are defined as follows:

      1.3.3.

      (i). An atomic formula is a formula.

      (ii). If φ and ψ are formulas, then (φ ψφ) are formulas.

      (iii). If υ is a variable and φ υ)φ is a formula.

      (iv). A sequence of symbols is a formula only if it can be shown to be a formula by a finite number of applications of (i)–(iii).

      , we may put is the least set T such that

      T contains all constant symbols and all variables υn, n = 0, 1, 2, …, and, whenever F is an m-placed function symbol and t1, …, tm T, then F(t1 … tmT.

      is the least set Φ such that

      every atomic formula belongs to Φ and, whenever φ, ψ Φ and υ is a variable, then (φ ψφυ)φ all belong to Φ.

      Note that we have tacitly used the letters t (with subscripts) to range over terms, υ to range over variables, and φ, ψ to range over formulas. Again, we emphasize that properties of terms and formulas of can only be established by an induction based on definitions 1.3.1 and 1.3.3.

      , →, ↔ as in (there exists) is introduced as an abbreviation defined as

      Some new conventions are the following:

      At this point we assume that the reader has enough experience in first-order predicate logic to continue the development on his own. In particular, we leave it to him to decide on the notions of subformulas, of free and bound occurrences of a variable in a formula, and to give a proper definition (based on definitions 1.3.1 and 1.3.3) of substitution of a term for a variable in a formula.

      We now come to an extremely important convention of notation. To make sure that the reader does not miss it, we enclose it in a box:

      We use t(υ0 … υn) to denote a term t whose variables form a subset of {υ0, …, υn}. Similarly, we use φ(υ0 … υn) to denote a formula φ whose free variables form a subset of {υ0, …, υn}.

      Note that we do not require that all of the variables υ0, …, υn be free variables of φ(υ0 … υn). In fact, φ(υ0 … υn) could even have no free variables. Also, we make no restriction on the bound variables. For example, each of the following formulas is of the form φ(υ0 υ1 υ2):

      A sentence is a formula with no free variables.

      and the other logical symbols listed. Such formulas are called identity formulas and they occur in every language. The following proposition is simple but important.

      PROPOSITION 1.3.4. The cardinal of the set of all formulas of is ||.

      To make all the above syntactical notions into a formal system we need logical axioms and rules of inferenceare divided into three groups.

      1.3.5. Sentential Axioms: Every formula φ which can be obtained from a tautology ψ for the sentence symbols of ψ . From now on we call such a formula φ a tautology .

      1.3.6. Quantifier Axioms:

      (i). If φ, ψ and υ is a variable not free in φ, then the formula

      is a logical axiom.

      (ii). If φ, ψ are formulas and ψ is obtained from φ by freely substituting each free occurrence of υ in φ by the term t (i.e., no variable x in t shall occur bound in ψ at the place where it is introduced), then the formula

      is a logical axiom.

      1.3.7. Identity Axioms: Suppose x, y are variables, t(υ0 … υn) is a term and φ(υ0 … υn) is an atomic formula. Then the formulas

      are logical axioms.

      There are two rules of inference.

      1.3.8. Rule of Detachment (or Modus Ponens): From φ and φ → ψ infer ψ.

      1.3.9. Rule of Generalization: From φ x)φ.

      Given the axioms and the rules of inference, we assume that the resulting notions of proof length of proof, theorem are already familiar to the reader. As we are dealing with the usual first-order logic with identity, we shall assume as known and make free use of all of the basic theorems and metatheorems of such formal systems.

      φ means that φ . If Σ , then Σ φ means that there is a proof of φ from the logical axioms and Σ. As the logical axioms are always assumed, we say that there is a proof of φ from Σ, or φ is deducible from Σ, whenever Σ φ. Σ is inconsistent can be deduced from Σ. Otherwise Σ is consistent. A sentence σ is consistent

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