We will consider Tarski's work in pure model theory and classical logic. His work in applied model theory—the model theory of various special theories—is discussed by Doner and van den Dries [1987], and McNulty [1986]. (However, the separation of “pure” and “applied” only becomes natural as the subjects mature; so we shall discuss applied model theory at least to some extent in Tarski's earlier work.)

Alfred Tarski (1901–1983) was awarded a Ph.D. in mathematics at Warsaw University in 1924. His teachers included the two leaders of the renowned Polish logic school, the logician-philosophers L. Leśniewski and J. Łukasiewicz. (Very soon Tarski was recognized as the third leader of the school.) Another teacher was the philosopher T. Kotarbiński, to whom Tarski dedicated his collected papers [56m]. Leśniewski was Tarski's thesis advisor; he transmitted to Tarski his interests in the metalanguage and in the theory of definition. Tarski's thesis ([23a], [24]) was about protothetic—the sentential calculus augmented by quantifiable variables ranging over truth functions. Its main result was that all the connectives are definable using only ↔ and ∀. By the same year, 1924, Tarski also had begun his prolific writings in set theory, and had discovered together with S. Banach, the leader of the Polish mathematicians, their famous “paradox” [24d] in measure theory. (For details see Lévy [1987].)

In 1927-29 Tarski held a seminar at Warsaw University on results he obtained in 1926-28. The seminar lay at the heart of what is now called model theory. The brief history of model theory up to that time had begun with the paper of L. Löwenheim [1915].

A distinctive feature of modern mathematics is the interaction between its various branches and the blurring of the boundaries between different areas. This is strikingly illustrated in the work of Alfred Tarski. He was a logician first and an algebraist second. His contributions to algebra can be divided into three (ill-defined and overlapping) categories, general algebra, the study of various algebraic structures arising from problems outside algebra, mostly in logic and set theory, and the use of concepts and techniques from logic in the study of algebraic structures. Even more roughly, these three categories could be labeled as pure algebra, applications of algebra to logic, and applications of logic to algebra.

Before Tarski came to the United States in 1939, he had written a series of papers on both the axiomatic and the structural aspects of Boolean algebras, and his inclination to algebraize mathematical problems is well illustrated by his paper [38g], Algebraische Fassung des Massproblems. Many of his later investigations of various types of algebraic structures are inspired by work done in this earlier period. However, beginning around 1940 there is a much greater emphasis on the study of algebra in its various aspects.

The paper [41], On the calculus of relations, is a landmark event in this respect. The object here was to find an axiomatic basis for the arithmetic of binary relations. The axioms that he chose are simple and natural (see Monk [1986]).

Alfred Tarski identified decidability within various logical formalisms as one of the principal themes for investigation in mathematical logic. This is evident already in the focus of the seminar he organized in Warsaw in 1926. Over the ensuing fifty-five years, Tarski put forth a steady stream of theorems concerning decidability, many with far-reaching consequences. Just as the work of the 1926 seminar reflected Tarski's profound early interest in decidability, so does his last work, A formalization of set theory without variables, a monograph written in collaboration with S. Givant [8−m]. An account of the Warsaw seminar can be found in Vaught [1986].

Tarski's work on decidability falls into four broad areas: elementary theories which are decidable, elementary theories which are undecidable, the undecidability of theories of various restricted kinds, and what might be called decision problems of the second degree. An account of Tarski's work with decidable elementary theories can be found in Doner and van den Dries [1987] and in Monk [1986] (for Boolean algebras). Vaught [1986] discusses Tarski's contributions to the method of quantifier elimination. Our principal concern here is Tarski's work in the remaining three areas.

We will say that a set of elementary sentences is a theory provided it is closed with respect to logical consequence and we will say that a theory is decidable or undecidable depending on whether it is a recursive or nonrecursive set. The notion of a theory may be restricted in a number of interesting ways. For example, an equational theory is just the set of all universal sentences, belonging to some elementary theory, whose quantifier-free parts are equations between terms.

One of the most extensive parts of Tarski's contributions to logic is his work on the algebraization of the subject. His work here involves Boolean algebras, relation algebras, cylindric algebras, Boolean algebras with operators, Brouwerian algebras, and closure algebras. The last two are less developed in his work, although his contributions are basic to other work in those subjects. At any rate, not being conversant with the latest developments in those fields, we shall concentrate on an exposition of Tarski's work in the first four areas, trying to put them in the perspective of present-day developments.

For useful comments, criticisms, and suggestions, the author is indebted to Steven Givant, Leon Henkin, Wilfrid Hodges, Bjarni Jónsson, Roger Lyndon, and Robert Vaught.

Tarski published his first geometry paper, [24b], in 1924. As is well known, the area of the union of two disjoint figures is the sum of the areas of these two figures. This observation is the basis of a method for proving that two figures, say A and B, have the same area: if we can divide each of the two figures A and B into a finite number of pairwise disjoint subfigures A1,…,An and B1,…,Bn such that for every i, figures Ai and Bi are congruent (we say that two such figures are equivalent by finite decomposition), then figures A and B have the same area. The method is by no means universal. For example a disc and a rectangle can never be equivalent by finite decomposition, even if they have the same area. Hilbert [1922, Kapitel IV] proved from his axiom system the so-called De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by a finite decomposition.

Hilbert's proof is elementary but difficult. In [24b] Tarski gave an easy but nonelementary proof of a stronger version of the De Zolt axiom:

If a polygon V is a proper subset of a polygon W then they are not equivalent by finite decomposition into any figures.

In this paper we prove a boundedness theorem in the theory ID1(W). This answers a question asked by Feferman, for example in [3]. The background is the following.

Let A[X, x] be an X-positive formula arithmetic in X. The theory ID1(PA) is an extension of Peano arithmetic PA by the following axioms:

for arbitrary formulas F; PA is a constant for the least fixed point of A[X, x]. Set-theoretically, PA can be defined by recursion on the ordinals as follows:

where is the first nonrecursive ordinal.

Now let a ≺ b be the arithmetic relation which expresses that the recursive tree coded by a is a proper subtree of the tree coded by b, and define

The least fixed point of Tree[X, x] is the set PTree of all well-founded recursive trees. We write W or Wα for PTree or , respectively. Since W is complete we have for all α < . If we define for each element a ∈ W its inductive norm ∣a∣ by ∣a∣≔ min{ξ: a ∈ Wξ}, then we have = {∣a∣: a ∈ W} and the elements of W can be used as codes for the ordinals less than .

Assume that B[X, x] is an X-positive formula arithmetic in X with the only free variables X and x, and assume that QB is a relation that satisfies

If we define

then we obviously have PB = IB.

We study the first order theory of function fields in the language of fields by using fundamental results on curves in algebraic geometry. We give some applications; for example, using a theorem of G. Cherlin, we prove the undecidability of function fields with nonzero characteristic over an algebraically closed field.

An explanation is given of why, after adding to a model M of ZFC first a Solovay real r and next a Cohen real c, in M[r] [c] a Cohen real over M[c] is produced. It is also shown that a Solovay algebra iterated with a Cohen algebra can be embedded into a Cohen algebra iterated with a Solovay algebra.

The following is a contribution to the abstract study of the model theory of modal logics. Historically, individual modal logics have been specified deductively; and, as a result, it has seemed natural to view modal logics as sets of sentences provable in some deductive system. This proof theoretic view has influenced the abstract study of modal logics. For example, Fine [1975] defines a modal logic to be any set of sentences in the modal language L□ which contains all tautologies, all instances of the schema (□(ϕ ⊃ Ψ) ⊃ (□ϕ ⊃ □Ψ)), and which is closed under modus ponens, necessitation and substitution.

Here, however, modal logics are equated with classes of “possible world” interpretations. “Worlds” are taken to be ordered pairs (a, λ), where a is a sentential interpretation and λ is an ordinal. Properties of the accessibility relation are identified with those classes of binary relational systems closed under isomorphisms. The origin of this approach is the study of alternative Kripke semantics for the normal modal logics (cf. Weaver [1973]). It is motivated by the desire that modal logics provide accounts of both logical truth and logical consequence (cf. Corcoran and Weaver [1969]) and the realization that there are alternative Kripke semantics for S4, B and M which give identical accounts of logical truth, but different accounts of logical consequence (cf. Weaver [1973]). It is shown that the Craig interpolation theorem holds for any modal logic which has characteristic modal axioms and whose associated class of binary relational systems is closed under subsystems and finite direct products. The argument uses a back and forth construction to establish a modal analogue of Robinson's theorem. The argument for the interpolation theorem from Robinson's theorem employs modal analogues of the Ehrenfeucht-Fraïssé characterization of elementary equivalence and Hintikka's distributive normal form, and is itself a modal analogue of a first order argument (cf. Weaver [1982]).

In his famous paper [1] on the elementary theory of finite fields Ax considered fields K with the property that every absolutely irreducible variety defined over K has K-rational points. These fields have been called pseudo algebraically closed (pac) and also regularly closed, and extensively studied by Jarden, Éršov, Fried, Wheeler and others, culminating with the basic works [8] and [11].

The above algebraic-geometric definition of pac fields can be put into the following equivalent model-theoretic version: K is existentially complete (ec) relative to the first order language of fields into each regular field extension of K. It has been this characterization of pac fields which the author extended in [2] to ordered fields. An ordered field (K, <) is called in [2] pseudo real closed (prc) if (K, <) is ec in every ordered field extension (L, <) with L regular over K. The concept of pre ordered field has also been introduced by McKenna in his thesis [15] by analogy with the original algebraic-geometric definition of pac fields.

Given a positive integer e, a system K = (K; P1, …, Pe), where K is a field and P1, …, Pe are orders of K (identified with the corresponding positive cones), is called an e-fold ordered field (e-field). In his thesis [9] van den Dries developed a model theory for e-fields. The main result proved in [9, Chapter II] states that the theory e-OF of e-fields is model con. panionable, and the models of the model companion e-OF are explicitly described.

The axioms for Zermelo-Fraenkel (ZF) set theory are an appealing but somewhat arbitrary-seeming assortment. A survey of the axioms does not suffice to reveal the source of their attraction. Accordingly, attempts have been made to ground ZF in principles whose appeal can be felt immediately. These attempts can be classified as follows. First, some of them propose to rest the ZF axioms directly on informal doctrine. The others propose to ground the ZF axioms in other formal axioms that can be regarded as more basic. When the latter approach is taken, ZF continues to draw on informal support, but the draft is made at a more basic level.

The same research can be classified in another way, according to the informal diagnosis offered for the paradoxes of set theory. In some cases, the diagnosis is that the paradoxical sets (such as the Russell set) fail to exist only because they would have to be too large; a set that would be sufficiently small must always exist. This is the doctrine of limitation of size. In other cases, the diagnosis is that the paradoxical sets fail to exist only because they would have to lie too high in a certain hierarchy of sets; a set that would lie sufficiently low in the hierarchy must always exist. This is the doctrine of the hierarchy. The present paper will investigate the latter doctrine, with the doctrine of size making a brief appearance at the end.

An attempt to answer the following question gave rise to the results of the present paper. Let be an arbitrary model of set theory. Does there exist an elementary extension of satisfying the two requirements: (1) contains an ordinal exceeding all the ordinals of ; (2) does not enlarge any (hyper) integer of ? Note that a trivial application of the ordinary compactness theorem produces a model satisfying condition (1); and an internal ultrapower modulo an internal ultrafilter produces a model satisfying condition (2) (but not (1), because of the axiom of replacement). Also, such a satisfying both conditions (1) and (2) exists if the external cofinality of the ordinals of is countable, since by [KM], would then have an elementary end extension.

Using a class of models constructed by M. Rubin using in [RS], and already employed in [E1], we prove that our question in general has a negative answer (see Theorem 2.3). This result generalizes the results of M. Kaufmann and the author (appearing respectively in [Ka] and [E1]) concerning models of set theory with no elementary end extensions.

In the course of the proof it was necessary to establish that all conservative extensions (see Definition 2.1) of models of ZF must be cofinal. This is in direct contrast with the case of Peano arithmetic where all conservative extensions are end extensional (as observed by Phillips in [Ph1]). This led the author to introduce two useful weakenings of the notion of a conservative end extension which, as shown by the “completeness” theorems in §3, can exist.

One of the typical preservation theorems in a first order classical predicate logic with equality L is the following theorem due to J. Łoś [4] and A. Tarski [9] (also cf. [1, p. 139]).

Theorem A (Łoś-Tarski). For any sentences A and B in L, the following two conditions (i) and (ii) are equivalent.

(i) Every extension of any model of A is a model of B.

(ii) The two sentences A ⊃ C and C ⊃ B are provable in L for some existential sentence C in L.

In [2], S. Feferman obtained a similar preservation theorem for outer extensions. In the following, we assume that L has a fixed binary predicate symbol <. Then Σ-formulas are formulas in L which are constructed from atomic formulas and their negations by applying ∧ (conjunctions), ∨ (disjunctions), ∀x < y (bounded universal quantifications), and ∃ (existential quantifications). An extension of an L-structure is said to be an outer extension of if ⊨ a < b and b ϵ ∣∣ imply a ϵ ∣∣ for any elements a, b in ∣∣.

Theorem B (Feferman). For any sentences A and B in L, the following two conditions (i) and (ii) are equivalent.

(i) Every outer extension of any model of A is a model of B.

(ii) The two sentences A ⊃ C and C ⊃ B are provable in L for some Σ-sentence C in L.

We show that relative to the consistency of a supercompact cardinal does not imply . The model-theoretic transfer property ⟨ℵ1, ℵ0⟩ → ⟨ℵω + 1, ℵω⟩ does not imply , and it is consistent to have an ultrafilter on ℵω + 1 which is λ-indecomposible for all ω < λ < ℵω.

All structures to be considered here have universe ω, and all languages come equipped with Gödel numberings. If is a structure, then D(), the open diagram of , can be thought of as a subset of ω, and it makes sense to talk about the Turing degree deg(D()). This depends on the presentation as well as the isomorphism type of .

For example, consider the ordering = (ω, <). For any B ⊆ ω, it is possible to code B in a copy of as follows: Let π be the permutation of ω such that for each n ∈ ω, π leaves 2n and 2n + 1 fixed if n ∈ B and switches 2n with 2n + 1 if n ∉ B. Let be the copy of such that ≃π. Then n ∈ B iff the sentence 2n < 2n + 1 is in D(). In §4, this idea will be used to show that for any structure that is not completely trivial, {deg(D()): ≃ } is closed upwards.

It would be satisfying to have a way of assigning Turing degrees to structures such that the degree assigned to a given structure measured the recursion-theoretic complexity of the isomorphism type and was independent of the presentation. Jockusch suggested the following.

This paper contains an example of a decidable theory which has

1) only a countable number of countable models (up to isomorphism);

2) a decidable saturated model; and

3) a countable homogeneous model that is not decidable.

By the results in [1] and [2], this can happen if and only if the set of types realized by the homogeneous model (the type spectrum of the model) is not .

If Γ and Σ are types of a theory T, define Γ ◁ Σ to mean that any model of T realizing Γ must realize Σ. In [3] a decidable theory is constructed that has only countably many countable models, only recursive types, but whose countable saturated model is not decidable. This is easy to do if the restriction on the number of countable models is lifted; the difficulty arises because the set of types must be recursively complex, and yet sufficiently related to control the number of countable models. In [3] the desired theory T is such that

is a linear order with order type ω*. Also, the set of complete types of T is not . The last feature ensures that the countable saturated model is not decidable; the first feature allows the number of countable models to be controlled.

In their paper [3], Hajnal and Komjáth define the following combinatorial principle:

Definition 1.1. Suppose κ is an infinite cardinal and n < ω. Then Hn(κ) is the statement: There is a function F: [κ]n → [[κ]ω]≤ω such that

(a) ∀A ∈[κ]n ∀Y ∈ F(A)(Y ⊆ min (A)), and

(b) .

Hn(κ) is related to a more general principle introduced by Hajnal and Nagy in [4]. For applications of these principles to free sets for set mappings and Ramsey games we refer the reader to [3] and [4].

In [3] Hajnal and Komjáth prove the consistency of ZFC + GCH + ∀n ∈ ω(Hn + 1(ωn + 1)), relative to an ω-Mahlo cardinal. They conjecture that L is a model of this theory, and suggest that the proof might require higher gap morasses. The first few cases of this conjecture are known to be true; it is easy to see that if CH holds then H1 (ω1) is true, and Laver proved that V = L implies H2(ω2). In this paper we go one step further and prove V = L → H3(ω3). Unfortunately our methods do not appear to give Hn (ωn) for n ≥ 4.

Most of our notation is standard. If X is any set and κ is a cardinal number then [X]κ is the set of subsets of X with cardinality κ, and [X]≤κ is the set of subsets of X with cardinality ≤ κ. If X is a set of ordinals then tp(X) is the order type of X.