二十世纪模型论发展迅猛，势不可挡

希尔伯特在“几何基础”中最初形成了数学模型的思想，数学进入新的发展轨道。

1954年，塔尔斯基悬宣布数学模型论正式成为现代数学的一个新分支，而且，发展很快。

2000年，英国Wilfrid Hodges发表文章，

列出231篇有关模型理论的研究论文，白纸黑字2说明了二十世纪模型论发展迅猛，而且势不可挡

在这个时期，鲁宾逊李彤模型论紧性定理证明。超实数系统（非准模型）的存在性。据此，J.Keisler精心撰写了“无穷小微积分”

教科书。这就是我的故事，

注：Wilfrid Hodges的文章（56页）在本文附件中。

袁萌陈启清7月4日

附件：

Model Theory (Draft 20 July 2000)

Wilfrid Hodges

1 The boundaries of the subject

In 1954 Alfred Tarski [210] announced that ‘a new branch of metamathematics’ had appeared under the name of the theory of models. The subject grew fast: the Omega Group bibliography of model theory in 1987 [148] ran to 617 pages. By the mid 1980s there were already too many dialects of model theory for anybody to be

expert in more than a fraction. For example very few model theorists could claim to understand both the work of Zilber and Hrushovski at the edge of algebraic geometry, and the studies by Immerman and Vardi of classiﬁcations of ﬁnite structures. And neither of these lines of research had much contact with what English-speaking philosophers and European computational linguists had come to refer to as ‘model-theoretic’ methods or concepts. Nevertheless all these brands of model theory had common origins and important family resemblances. Some other things called models deﬁnitely lie outside the family. For example this chapter has nothing to say about ‘modelling’, which means constructing a formal theory to describe or explain some phenomena. Likewise in cognitive science the ‘mental models’ of Gentner and Stephens [67] or Johnson-Laird [100] lie outside our topic. Alltheﬂavoursofmodeltheoryrestononefundamentalnotion, andthat is the notion of a formula φ being true under an interpretation I. The classic treatment is Tarski’s paper [202] from 1933. In this paper Tarski supposes that we have a language L with a precisely deﬁned syntax. Ignoring punctuation, the symbols of L are of two kinds: constants and variables. The constants have ﬁxed meanings; they will usually include logical expressions such as ‘and’ and ‘equals’. The variables have no meaning, but (to shortcircuit Tarski’s very careful formulation a little) we can assign an object to each variable, and ask whether a given formula φ of L becomes true when each variable is regarded as a name of its assigned object. The grammatical categories of the variables determine what kinds of object can be assigned to them; for example we can assign individuals to individual variables, classes

of individuals to class variables, and so on. If A is an allowed assignment of objects to variables and A makes φ true, then A is said to satisfy φ, and to be a model of φ, and φ is said to be true in A. (In [202] Tarski says ‘satisfy’, and moves on to give a deﬁnition of ‘true’ in terms of ‘satisfy’.) One of Tarski’s main aims in this paper was to show that for certain kinds of language L, the relation ‘A satisﬁes φ’ is deﬁnable using only set theory, the syntax of L and the notions expressed by the constants of L. Thus one speaks of Tarski’s deﬁnition of truth (or of satisfaction). In several papers around 1970, Tarski’s student Richard Montague [138] set out to show that Tarski’s treatment applies to some nontrivial fragments of English. This work of Montague is a paradigm of what philosophers and linguists call model-theoretic semantics. Tarski himself ([202] §6 end) foresaw this development, but he suspected that it could only be carried through by rationalising natural language to such an extent that it might not ‘preserve its naturalness and [would] rather take on the characteristic features of the formalized languages’. For the rest of this chapter, we shall only be concerned with formulas of artiﬁcial languages. Tarski’s 1933 paper brought into focus a number of ideas that were in circulation earlier. The notion of an assignment satisfying a formula is implicit in George Peacock ([151], 1834) and explicit in George Boole ([25] p. 3, 1847), though without a precise notion of ‘formula’ in either case. The word ‘satisfy’ in this context may be due to Edward V. Huntington (for example in [97] 1902). Geometers had spoken of gypsum or paper ‘models’ of geometrical axioms since the 17th century; abstract ‘models’ appeared during the 1920s in writings of the Hilbert school (von Neumann [147] 1925, Fraenkel [59] p. 342, 1928). In 1932 Kurt G¨odel wrote to Rudolf Carnap that he was intending to publish ‘eine Deﬁnition f¨ur ‘wahr’ ’ ([73]). He never published it. We know that by 1931 G¨odel already had a good understanding of deﬁnability of truth in systems of arithmetic (see Feferman [57]); there is no solid evidence on whether he had thought about general set-theoretic deﬁnitions of truth before Tarski’s paper was published.

2 One structure at a time

In Tarski’s paper [202] he gives four examples of truth deﬁnitions, one of which (in his§3) he describes as ‘purely accidental’ because it doesn’t follow the general strategy of his main deﬁnition. In this example he considers what today we would call the structure A of all subsets of a given set a, with

relation ⊆; he discusses what can be said about A using the corresponding ﬁrst-orderlanguage L. (Thismaybeananachronism; onecouldalsodescribe his example as the structure consisting of the set a with no relations, and a corresponding monadic second-order language.) Tarski works out an explicit deﬁnition of the relation ‘φ is true in A’, where φ ranges over the sentences of L. To bring this example within his scheme, Tarski counts the symbol ⊆ as a constant; the two quantiﬁers are constants too, understood as ranging over the set of subsets of a. Tarski doesn’t call A a structure—in fact he doesn’t even have a symbol for it—but the following features make it an example of what later became known as a structure. It carries a domain or universe (the set of subsets of a); it also carries a binary relation on the domain, labelled by the symbol ‘⊆’. A structure can also carry labelled relations and functions of any ﬁnite arity on the domain, and labelled elements of the domain. Structures in this sense are an invention of the second half of the 19th century—for example Hilbert handled them freely in his Foundations of Geometry ([85] 1899). Richard Dedekind ([44] 1871 and [39] 1872) frequently used the nameSystem for structures (and also for sets—apparently he thought of a structure as a set that comes with added features). Weber and Hilbert spoke of ‘Systeme von Dingen’, to distinguish from axiom systems. In model theory the name ‘system’ persisted until it was replaced by ‘structure’ in the 1950s, it seems under the inﬂuence of Abraham Robinson [162] and Bourbaki [28]. (Augustus De Morgan introduced the term ‘universe’ in 1846 [40], to mark the fact that one deals with a determinate set of things which may be diﬀerent in diﬀerent discourses. ‘Gebiet’—which is ‘domain’ in German—is a loose term, but its use in structures may owe something to Hermann Grassmann [72] 1844; Dedekind used it as a synonym for ‘System’.) Tarski’schoiceofexamplewasnotanaccident. LeopoldL¨owenheim[123] §4 (1915) had already studied the same example within the context of the Pierce-Schr¨oder calculus of relatives, and he had proved a very suggestive result. In modern terms, L¨owenheim had shown that there is a set of ‘basic’ formulas of the language L with the property that every formula φ of L can be reduced to a boolean combination ψ of basic formulas which is equivalent to φ in the sense that exactly the same assignments to variables satisfy it in A. Thoralf Skolem [191] §4, 1919 and Heinrich Behmann [18] 1922 had reworked L¨owenheim’s argument so as to replace the calculus of relatives by more modern logical languages. In 1927 C. H. Langford [114], [115] applied the same ideas to dense or discrete linear orderings. Tarski realised that not only the arguments of L¨owenheim and Skolem, but also the heuristics behind them, provided a general method for analysing

structures. This method became known as the method of quantiﬁer elimination. In his Warsaw seminar, starting in 1927, Tarski and his students applied it to a wide range of interesting structures. Important early examples were the ﬁeld of real numbers (Tarski [201] 1931) and the set of natural numbers with symbols for 0, 1 and addition (Presburger [160] 1930). In both these cases the method yielded (a) a small and easily described set of basic formulas, (b) a description of all the relations deﬁnable in the structure by ﬁrst-order formulas, (c) an axiomatisation of the set of all ﬁrst-order sentences true in the structure and (d) an algorithm for testing the truth of any sentence in the structure. (Here (b) comes at once from (a). For (c), one would write down any axioms needed to reduce all formulas to boolean combinations of basic formulas, and all axioms needed to determine the truth of basic sentences. Then (d) follows since the procedure for reducing to basic formulas is eﬀective.) One should note two characteristic features of the method of quantiﬁer elimination. First, the structure serves only to provide a stock of axioms. True, nontrivial theorems about the structure may be useful for ﬁnding the axioms. (For the ﬁeld of real numbers, Tarski used Sturm’s theorem.) But once the axioms are on the table, the rest of the work (and it can be heavy) is entirely syntactic and unlikely to appeal to model theorists. Tarski was aware of this. As late as 1978 he was defending the method of quantiﬁer elimination against modern methods which often prove more eﬃcient. ... It seems to us that the elimination of quantiﬁers, whenever it is applicable to a theory, provides us with direct and clear insight into both the syntactical structure and the semantical contents of that theory—indeed, a more direct and clearer insight than the modern more powerful methods to which we referred above. (Doner, Mostowski and Tarski [43] p. 1f.) Second, the method works on just one structure at a time. It involves no comparison of structures. Let me dwell on this for a moment. In 1936 [203] Tarski reworked and reﬁned Langford’s results on discrete linear orderings. Starting from an arbitrary discrete linear ordering A, he would follow the method until he came to a choice of axiom that was true in some discrete linear orderings and false in others. He found that he needed only the following axioms or their negations: There is a ﬁrst element. There is a last element. There are at least n elements (n a positive integer).

From this he could read oﬀ exactly what are the possible completions of the theory of discrete linear orderings. In this way he got information not about a single structure but about a class of structures. (In the Appendix to [203], Tarski tells us that by 1930 he had deﬁned the relation of elementary equivalence, in modern symbols ≡: A ≡ B if the same ﬁrst-order sentences are true in A as in B.) Nevertheless the quantiﬁer elimination itself involves only one structure at a time, and any comparisons come later. There are two common misconceptions about the method of quantiﬁer elimination. The ﬁrst is that the basic formulas should be quantiﬁer-free. This was never the intention, though it happened in some cases. (The name means that we reach the basic formulas by eliminating quantiﬁers, not that all quantiﬁers are removed.) Later model theorists said that a theory has the property of quantiﬁer elimination if every formula is equivalent, in all models of the theory, to a quantiﬁer-free formula with the same free variables. One must avoid confusing the property with the method. The second misconception is that the main aim of the method was to solve decision problems. This is about half true. It was Behmann’s paper [18] in 1922 that introduced the term ‘Entscheidungsproblem’ (cf. Mancosu [135]), long before Church and Turing in 1935–6 made the notion of an eﬀectivelydecidablesetprecise. AlsoSkolemin[191]presentedhisquantiﬁer elimination as a way of ‘evaluating’ (auswerten) formulas; and in [197] he emphasised decidability (though he had other aims too). But Langford’s two papers [114], [115] show no awareness of decidability; he is concerned only with ﬁnding complete sets of axioms. Tarski’s reworkings of the quantiﬁer eliminations of Langford [203] and Skolem [204] are silent about decidability. Tarski [201] in 1931 and his student Presburger [160] in 1930 mention in passing that a decision procedure falls out of the quantiﬁer elimination. Not until 1940 does Tarski come out with the following (not very modeltheoretic) statement: It is possible to defend the standpoint that in all cases in which a theory is tested with respect to its completeness the essence of the problem is not in the mere proof of completeness but in giving a decision procedure (or in the demonstration that it is impossible to give such a procedure). ([206] Note 11, published in 1967 from the 1940 page proofs). By contrast it always was one of the main aims of quantiﬁer elimination, in both Skolem and Tarski, to answer the question ‘What relations are deﬁnable in a given structure A by means of formulas of a given formal language?’. (In 1910 Hermann Weyl [222] had introduced the class of

ﬁrst-order deﬁnable relations over a relational structure, but without using a formal language.) This remained a central question of model theory in all periods. For example, when the structure A is an algebraically closed ﬁeld, we are asking what are the constructible sets of aﬃne geometries over A. It took some time for model theorists to realise that this gave them an entry into algebraic geometry. Abraham Robinson realised it after a fashion, and with Peter Roquette he gave a model-theoretic proof of the ﬁniteness theorem of SiegelandMahler[170]. Thiswaspublishedin1975, shortlyafterRobinson’s premature death. But the real breakthrough came with the discovery of Morley rank in algebraically closed ﬁelds and the subsequent development of geometric model theory. Already Boris Zilber’s 1977 paper [225] is largely about ﬁrst-order deﬁnable relations. Likewise it was a long time before model theorists realised what a strong tool they had in Tarski’s description of the ﬁrst-order deﬁnable sets in the ﬁeld of real numbers. These sets are precisely the unions of ﬁnitely many sets, each of which is either a singleton or an open interval with endpoints either in the ﬁeld or ±∞; in 1986 Anand Pillay and Charles Steinhorn [157] introduced the name o-minimal for ordered structures in which these are all the ﬁrst-order deﬁnable sets. Lou van den Dries had pointed out in 1984 [45] that the o-minimality of the ﬁeld of real numbers already gives strong information about deﬁnable relations of higher arity—in particular it allows one to recover a form of cell decomposition in the style known to geometers. Julia Knight, Pillay and Steinhorn [109] generalised this cell decomposition to all o-minimal structures. One of the outstanding results in this area of model theory was Alex Wilkie’s proof in 1991 [223] that the ﬁeld of real numbers with the exponentiation function added is o-minimal and has model-complete theory (i.e. the existential formulas form a basic set). Wilkie’s method was quite diﬀerent from Tarski’s; the problem may be beyond the reach of methods as eﬀective as Tarski’s. But Wilkie could call on the advances in model theory made by Abraham Robinson and his generation, and also a substantial body of work on real-algebraic geometry by Khovanskii and others.

3 The metamathematics of ﬁrst-order classes of structures

Strictly speaking, if one compares the sentences true in two structures A and B, one is no longer using the symbols of the structures as constants. This

would have made it awkward to apply Tarski’s [202] truth deﬁnition directly within model theory. But Tarski was aware that there was a problem here, and in [205] §37 (1941 edition AND THE POLISH??) he proposed a way of dealing with it. In a typical axiom system there are (again ignoring punctuation) not two but three kinds of symbol. First come the logical symbols for ‘and’, ‘or’ and so forth. Second there are variables. But thirdly there are what Tarski calls ‘primitive terms’. These include the relation and function symbols of the axiom system, and in general they also include a term standing for the domain over which quantiﬁers are to range. The primitive terms have no meaning, so they are unsuitable to appear as constants in the 1933 truth deﬁnition. Therefore Tarski applies that deﬁnition to a sentence φ and a structure A indirectly: he ﬁrst forms an expression φ by replacing the primitive terms in φ by variables, and then he says that A is a model of φ if the assignment of the appropriate features of A to the corresponding variables in φ satisﬁes φ. This is strangely roundabout. In a diﬀerent way, so is Tarski’s procedure of [208], where he replaces formulas by functions from structures to deﬁnable sets. But it must have become clear by 1950 that the truth deﬁnition should be rewritten to handle languages with three levels of symbol—logical, variable, primitive. Tarski’s paper [210] already assumes such a truth definition, and Tarski and Robert Vaught wrote out the details in 1957 [213]. (Mostowski [143] §3 1952 is an anticipation but in a diﬀerent idiom.) As urged by C. S. Peirce [154] in 1884, Tarski and Vaught took the quantiﬁers to range over the domain implicitly, so that no primitive symbol for the domain was needed. They referred to the remaining primitive symbols as the non-logical constants. So now people had a direct model-theoretic deﬁnition of ‘structure A is a model of sentence φ’, and they could straightforwardly deﬁne the class Mod(T) of all structures which are models of a given set T of ﬁrst-order sentences. In this notation, Tarski’s deﬁnition ([204] p. 8, 1953) of the relation ‘sentence φ is a logical consequence of theory T’ becomes Mod(T)⊆Mod({φ}) i.e. ‘Every model of T is a model of φ’. Changes of viewpoint usually generate new terminology. In 1950 there was still no agreed name for a set of sentences of a formal language. For example one ﬁnds them referred to as Axiomsysteme, sets of axioms, sets of postulates, sets of laws, sets of propositions, and less often Theorien (e.g. in Hilbert [86]). In 1935/6 Tarski’s ‘deductive theories’ were sets of expressions

with a logic attached ([203]). During the mid 1950s the word theory came to be accepted as the standard word for a set of sentences of a logical language. (But not by Robinson, who continued to write of ‘sets of axioms’ or ‘sets of sentences’ throughout his career.) A number of earlier metamathematical results fell into place at once within this general picture.

(a) In 1906 Gottlob Frege [64] attacked the idea of using a set of formal axioms to deﬁne a class of structures. One thing that he found particularly scandalous was that mathematicians should in good faith use the same symbol to mean diﬀerent things (in diﬀerent structures) within one and the same discourse: In der Tat, wenn es sich darum handelte, sich und andere zu t¨ auschen, so g¨abe es kein besseres Mittle dazu, als vieldeutige Zeichen. (Indeed, if it were a matter of deceiving oneself and others, there would be no better means than ambiguous signs. [64] p. 307.) But in the new truth deﬁnition the relation ‘symbol S names relation R in structure A’ has an unambiguous and purely mathematical content, so that even the most punctilious model theorist can use it with a pure heart. (Frege had other objections too, and they need other answers.)

(b) In the years around 1900, Giuseppe Peano ([151] 1891), Hilbert ([85] 1899), Huntington ([97] 1902) and their colleagues considered a number of questions of the form: Is axiom φ in the axiom set T deducible from the other axioms in T? (Hilbert’s cited work includes the famous example of Euclid’s parallel postulate.) To show the answer No, one proves that some model of T\{φ}is not a model of φ, and one usually does this by describing such a model explicitly. The truth deﬁnition allows us to give a purely mathematical explanation of what is happening in arguments of this kind. Iaddawordofcautionhere. In1936[204]Tarskiproposedadeﬁnitionof ‘logical consequence’ between sentences without non-logical constants. The notion that he called ‘logical consequence’ in 1953 [212] was the natural analogue for languages with three levels of symbol, but since it applies to a diﬀerent class of languages, it can hardly pick up the same relation. Mathematical logicians who use Tarski’s 1953 terminology are sometimes said by philosophers to be endorsing ‘the model-theoretic theory of logical consequence’. This seems to be a case of cross purposes; mathematical logicians who use the phrase are simply referring to a useful set-theoretical notion,

not taking sides in a contentious conceptual analysis. The description ‘model-theoretic theory’ does certainly apply to Tarski’s 1936 [204] proposal, in the same broad sense that Tarski’s 1933 truth deﬁnition is model-theoretic. Tarski also used the word ‘model’ in the 1936 paper, but (uncharacteristically) to stand for a reinterpretation of terms that already carry a meaning. His proposal was that a sentence φ should count as logically true if and only if every model (in this odd sense) of φ is true. He noted that to make this deﬁnition work, one would need to restrict the class of terms that can be reinterpreted, so that in some appropriate sense the reinterpreted sentence has the same ‘form’ as the original. (And for logical consequence he proposed a similar deﬁnition.) Part of Tarski’s 1936 proposal was several hundred years old; already in the twelfth century Peter Abelard [1] p. 255 had noted that a ‘perfect’ syllogism remains valid when its terms are altered. Unknown to Tarski, Bernard Bolzano [24] §§147f in 1837 had suggested using this idea as a deﬁnition of analyticity: in Bolzano’s terminology a true proposition P is logically analytic if and only if we get a true proposition whenever we vary the non-logical ideas that appear in P. Tarski’s 1936 proposal adds one further ingredient that is missing in Bolzano: in the reinterpretation, terms should be allowed to stand for any objects of suitable type in the universe, regardless of whether they already have names in the language. This was a byproduct of the truth deﬁnition, but for Tarski it was one of the most important features of his proposal.

(c) In his paper of 1915 on the calculus of relatives [123], L¨owenheim showed that every sentence of ﬁrst-order logic, if it has a model, has a model with at most countably many elements. His proof has several interesting features, including his introduction of function symbols to reduce the satisﬁability of a sentence ∀x∃yφ (x,y) to the satisﬁability of the sentence ∀xφ (x,F(x)). Thus it seems that L¨owenheim invented Skolem functions, if we forgive him his bizarre explanation of the passage from the ﬁrst sentence to the second. (He uses inﬁnite, possibly uncountable, strings of existential quantiﬁers!) Skolem tidied up L¨owenheim’s argument and strengthened the result in two alternative ways. In 1920 [192] he showed, using the axiom of choice and a coherent account of Skolem functions, that if T is a countable ﬁrstorder theory (in fact he allows countable conjunctions and disjunctions too,

and inﬁnite quantiﬁer strings) with a model A, then T has a model B with at most countably elements; the proof shows that B can be taken as an elementary substructure of A, but at this date Skolem lacked even the notion of substructure. Two years later [193] he showed without the axiom of choice that every countable ﬁrst-order theory with a model has an at most countable model. He used this result to construct a countable (and hence nonstandard) model of Zermelo-Fraenkel set theory. These results of Skolem seem to have attracted little attention before the 1950s. But from the new perspective of the 1950s they were central results about classes of the form Mod(T). In 1957 Tarski and Vaught [213] proved the stronger result that if L is a ﬁrst-order language with at most λ formulas, and A is a structure for L of cardinality >λ, then A has an elementary substructure of cardinality λ. This result was known as the Downward L¨owenheim-Skolem-Tarski Theorem (though it soon became usual to drop Tarski’s name from the list in the interests of brevity).

(d) In his 1930 PhD thesis [68], Kurt G¨odel reworked Skolem’s argument from [193] and used it to show that if T is any syntactically consistent theory in a countable ﬁrst-order language, then relations and functions can be deﬁned on the natural numbers, corresponding to the relation and function symbols of T, so that the deﬁned relations and functions make the natural numbers N into a model of T (perhaps with an equivalence relation for equality). Thus one could build a structure by ‘interpreting’ the language of the structure in another given structure. Mostowski [142] 1948, Tarski [212] 1953, Ershov [56] 1974 and others developed this idea, usually under the name of interpretation. Since their aim was to prove undecidability results, they thought in terms of the syntactic interpretation of the theory T rather than in terms of building a new structure. But when stability theorists found they needed a notion of one structure being interpretable in another, Ershov’s notion was the one they used. Shelah [185] (Chapter III §6) described how one might think of the elements of structures interpretable in a structure M as imaginary elements of M. If the theory T in G¨odel’s argument was itself arithmetically deﬁnable, then one could place bounds on the arithmetical complexity of the interpreting relations and functions. So G¨odel’s paper was a ﬁrst step towards eﬀective model theory. Various people studied eﬀective models at various times and places—the text of Goncharov and Ershov [71] has nearly 400 references—but it never became a mainstream topic.

(e) It was natural to look for upward versions of Skolem’s results. Tarski claimed in 1934 (in a note added by the editors to the end of Skolem [195]) that in 1927/8 he had proved that every consistent ﬁrst-order theory with no ﬁnite model has a model with uncountably many elements. (Vaught [214] p. 160 reports the few facts that are known about this early proof by Tarski.) Anatolii Mal’tsev [130] 1936 stated that every ﬁrst-order theory with an inﬁnite model has models of ‘all cardinalities’ (presumably he meant all high enough cardinalities). His proof rested on the Compactness Theorem, on which see (f) below. Given the Compactness Theorem, it is easy to prove that if L is a ﬁrst-order language with at most λ formulas, and A is a structure for L of cardinality <λ, then A has an elementary extension of cardinality λ. This result became known as the Upward L¨owenheimSkolem-Tarski Theorem—though again Tarski’s name was often dropped. The irony was that it was Skolem [196], not Tarski, who refused to accept that the theorem was true (though he allowed that it might be deducible within some formal set theories).

(f) The Compactness Theorem for ﬁrst-order logic states that if T is a ﬁrst-order theory such that every ﬁnite subset of T has a model, then T has a model. Results equivalent to this appear in papers of G¨odel (i) for countable languages, [68] in 1930 and (ii) for propositional languages of arbitrarily cardinality, [69] in 1932. In 1941 Mal’tsev [132] stated the full theorem, referring to his paper [130] of 1936 for the proof. In fact [130] doesn’t state the theorem, though it does contain a correct proof in two parts. First Mal’tsev states and proves the Compactness Theorem for a propositional language of arbitrary cardinality. Then he shows how this theorem can be lifted to a theory S in an arbitrary ﬁrst-order language, by ﬁrst passing to Skolem normal form and then introducing individual constants as ‘witnesses’ (to use the modern jargon) in order to replace S by a propositional theory—let us call it T—with the properties: (i) If every ﬁnite subset of S is satisﬁable then every ﬁnite subset of T is satisﬁable, (ii) if T is satisﬁable then S is satisﬁable. (i) is routine to check from Mal’tsev’s construction, but he never states it explicitly in the paper; this is hardly a ground for denying him the theorem. The familiar proof of the Compactness Theorem by way of a completeness proof for a proof calculus for ﬁrst-order languages of arbitrary cardinality is a slight revision (due to Gisbert Hasenjaeger [78]) of the proof given independently by Leon Henkin [81] in 1949, from his PhD thesis. The detour through a proof calculus is very easily avoided by replacing ‘T is syntactically consistent’ throughout the argument by ‘All ﬁnite subsets of T have

models’. Any proof of the Compactness Theorem has to use some method for building structures of arbitrarily high cardinality. Interpretation in N, as in G¨odel’s argument, is no longer an option. Both Mal’tsev and Henkin take the elements of their models to be individual constants (or equivalence classes of these constants, to give ‘=’ its standard meaning). Since there are uncountably many of them, these constants can only be linguistic objects in an abstract sense. In fact the central point seems to be that both Mal’tsev and Henkin introduced a well-ordered sequence of objects and then used it as a template to build the model around. Constructions of this kind were popular in the 1970s, particularly in connection with Jensen’s prediction principles (such as diamond) that allow one to ‘predict’ how the model will sit around the template; see for example Shelah on the Whitehead problem [184] 1974. Frayne, Morel and Scott [63] 1962 discovered another proof of the Compactness Theorem along completely diﬀerent lines. They used ultraproducts (on which see the next section), after Tarski had noticed that reduced products can be used to prove compactness for sets of Horn sentences. The name ‘Compactness Theorem’ is from Tarski’s 1950 Congress address, [208]. Until the late 1940s the Compactness Theorem went almost unnoticed in the West. This shouldn’t be a surprise; at ﬁrst sight the Compactness Theorem has nothing useful to say about the kinds of question we considered in section 2 above. But as soon as attention shifted to axiomatically deﬁned classes, it became clear that the Compactness Theorem was a powerful tool for building structures within such a class. The credit for this realisation goes to Mal’tsev and independently Abraham Robinson. We have already notedthat Mal’tsev [130] used compactness to derive a form of upward L¨owenheim-Skolem theorem. More startling, he used compactness in two group-theoretic papers [131] 1940 (implicitly) and [132] 1941 (explicitly) to prove local theorems in group theory. The result of [131] is the now well-known theorem that if all the ﬁnitely generated subgroups of a group G have faithful linear representations of degree n over ﬁelds, then so does G. These were the ﬁrst examples of results interesting to mathematicians in other ﬁelds, but proved by model theory. Unfortunately the ﬁrst notice of [132] in the West seems to have been in 1959 ([134]). But meanwhile Robinson had independently proved the Compactness Theorem in his PhD thesis [163] and had used it to deduce several mathematical statements such as the following ([162] p. 7, which I state in more modern terms): ‘Every purely transcendental extension K of the ﬁeld R of rational numbers has a ﬁeld extension which is an elementary extension of R and

adds no new element algebraic over the transcendence basis of K.’ Section 5 will examine how Robinson developed this line of work.

4 Expressive power

As soon as one has in place a serviceable deﬁnition of the relation ‘Structure A is a model of ﬁrst-order sentence φ’, certain things become almost routine. One can deﬁne the class Mod(T) of all models of a theory; dually one can deﬁne the set Th(K) of all ﬁrst-order sentences true throughout the class K of structures. Together Mod and Th form a Galois connection. A class of the form Mod(T) is said to be an EC class or an elementary class if T is ﬁnite, and an EC∆ class or an elementary class in the wider sense in the general case. Thus Tarski [210] in 1954, except that he had ‘arithmetical’ for ‘elementary’. An obvious question to ask is: (1) Given a ﬁrst-order theory T, what can one say in general about its class of models? For example, are there interesting general constructions that will give models of T with interesting properties? The dual question would be: Given a class of structures, what can one say in general about the set of sentences true in all of them? But this is a boring question (or at least it was until computer scientists found themselves looking for eﬃcient ways to determine, for a given structure A and sentence φ, whether φ is true in A). So instead one asked: (2) Given two structures A and B, how can one determine whether the same sentences are true in both of them? These are technical questions calling for technical answers. During the 1950s and early 1960s a stream of answers emerged. The text of Chang and Keisler [33], published in 1973, is a compendium of the main achievements. There were two other trends in the model theory of this period, neither of themmuchconnectedwiththequestions(1)and(2), thoughtheyhadalarge inﬂuence on the ways in which the technical breakthroughs were exploited. The ﬁrst trend was that people began to take a close interest in languages which are stronger than ﬁrst-order but more tractable than second-order, for example inﬁnitary languages and languages with cardinality quantiﬁers (see section 9 below). So when someone had introduced a technique for ﬁrstorder languages, he or she could move on to testing the same technique on stronger and stronger languages. Often a variant of the technique would still work, but set theoretic assumptions and arguments would begin to appear. An observation of William Hanf [76] helped to organise this area: he noted that for any reasonable language L there is a least cardinal κ (which became

known as the Hanf number of L) such that if a sentence of L has a model of cardinality at least κ then it has arbitrarily large models. A great deal of work and ingenuity went into ﬁnding the Hanf numbers of a range of languages. One eﬀect of this trend was that the centre of gravity of research on questions (1) and (2) moved steadily away from ﬁrst-order languages and towards inﬁnitary languages during the period from 1950 to 1970, bringing a heady dose of set theory into the subject. Allow me two anecdotes. In about 1970 a Polish logician reported that a senior colleague of his had advised him not to publish a textbook on ﬁrst-order model theory, because the subject was dead. And in 1966 David Park, who had just completed a PhD in ﬁrst-order model theory with Hartley Rogers at MIT, visited the research group in Oxford and urged us to get out of ﬁrst-order model theory because it no longer had any interesting questions. (Shortly afterwards he set up in computer science.) The second trend was that this was the period in which model theorists began to take maps between structures seriously (see section 5). The eﬀects of this trend were both subtler and more profound. Most of the rest of this chapter will be devoted to them. Here follow three of the answers to (1).

(1a) Ultraproducts. In 1955 Jerzy SLo´s [121] described a construction which combined a family of structures by means of an ultraﬁlter D on the power set algebra of the index set. He stated that a ﬁrst-order sentence φ is true in the constructed structure if and only if the set of indices at which φ is true lies in D. This became known as #Lo´s’s Theorem, and the construction itself gained the name of ultraproduct; an ultraproduct of isomorphic structures was called an ultrapower. (It came to light that ultraproducts or their close relatives had been used earlier by Skolem [194] 1934, Hewitt [84] 1948 and Arrow [3] 1950. Skolem’s application was model-theoretic, to build a structure elementarily equivalent to the natural numbers with + and·but not isomorphic to them.) We noted in the previous section that one can prove the Compactness Theorem directly by ultraproducts. Ultrapowers also give a direct construction of arbitrarily large elementary extensions of an inﬁnite structure. In 1961 Dana Scott [177] used a measurable cardinal to construct an inner model of set theory that was an ultrapower of the universe. This ensured that ultraproducts remained an essential tool of set theory long after most model theorists had lost interest in them. The further development of ultraproducts within model theory largely

revolved around what one might call their injective-like properties: various commutative diagrams could be completed by adding a mapping from some given structure to an ultraproduct of other given structures. The simplest example was Frayne’s Lemma ([63] 1962): If A is elementarily equivalent to B then A is elementarily embeddable in some ultrapower C of B. Frayne’s proof of this result constructed the embedding and the ultrapower C simultaneously. Jerome Keisler reorganised arguments of this kind in a style that was to prove important—Morley and Vaught [141] speak of Keisler’s ‘ “one element at a time” property’. Keisler himself (footnote on Theorem 2.2 of his doctoral dissertation, [104] 1961) compared his procedure with the elementat-a-time methods used by Cantor [30] and Hausdorﬀ [79] to build up isomorphismsbetweendenselyorderedsets. Frayne’sLemmaisagoodexample to illustrate the point. Let the sequence (ai : i<κ) list all the elements of A. The aim then is to construct, by induction on α ≤ κ, a sequence (ci : i<α) of elements of the ultrapower C so that the map ai

→ ci is a partial elementary map, i.e. any formula satisﬁed by the ai in A is also satisﬁed by the corresponding ci in C. We can construct such a sequence provided that we know that C has the following property: Given any set X of fewer than κ elements of C and any complete 1-type over these elements (i.e. set of formulas φ(x) with elements of X as parameters, which is maximal consistent with the theory of C), there is an element of C realising this type. Keisler said that if C had (essentially) this property, then it was κ-replete, and he then showed how to construct κ-replete ultrapowers. Keisler’s definition was slightly inconvenient—he counted the formulas rather than the parameters—and a corrected version took the name κ-saturation. A structure of cardinality κ was called saturated if it was κ-saturated. (The name is from Vaught [215] in the case κ = ω.) Thus it turned out that ultraproducts were useful largely because of their high saturation. Highly saturated elementary extensions are easy to construct directly by realising types and taking unions of elementary chains. Since saturation is a simpler concept than ultraproducts, ultraproducts largely dropped out of use within the model theory community once the early enthusiasm had worn oﬀ. But there remain some important theorems for which ultraproducts give the only known reasonable proofs; one is Keisler’s theorem [106] that uncountably categorical theories fail to have the ﬁnite cover property. Many algebraists still value ultraproducts for their transparency.

(1b) Omitting types. Ultraproducts give highly saturated models. One

also wanted a way of building models that are very unsaturated. In 1959 Vaught [215] gave the classic omitting types theorem for countable models of complete ﬁrst-order theories. This theorem allows one to omit countably many types at once; Vaught attributes this feature to Ehrenfeucht. The paper also contains Vaught’s Conjecture as a question: ‘Can it be proved, without the use of the continuum hypothesis, that there exists a complete theory having exactly ℵ1 non-isomorphic denumerable models?’ (The Conjecture is that there is no such theory. It is still open in spite of some impressive work by Rubin, Steel, Buechler, Newelski and others on special cases.) There were several close variants of omitting types. The Henkin-Orey theorem [149] was one that appeared before Vaught’s paper, while Robinson’s ﬁnite forcing [14] and Grilliot’s theorem [74] on constructing families of models with few types in common were two that came later. Martin Ziegler [224] made ﬁnite forcing more palatable by recasting it in terms of Banach-Mazur games; the same recasting works for all versions of omitting types. Finite forcing builds existentially closed models; these were introduced into model theory by Michael Rabin [161] 1962 and Per Lindstr¨om [119] 1964. During the 1970s Belegradek, Ziegler, Shelah and others put a good deal of energy went into constructing existentially closed groups, after Macintyre [125] 1972 had shown that they have remarkable deﬁnability properties.

(1c) Indiscernibles. In 1956 Ehrenfeucht and Mostowski [50] showed, using Ramsey’s Theorem, that if T is a complete ﬁrst-order theory with inﬁnite models and (X,<) is a linearly ordered set, then T has a model A whose domain includes X, and for each ﬁnite n, any two strictly increasing n-tuples from X satisfy the same formulas in A. (In short, (X,<) is an indiscernible sequence in A—though Ehrenfeucht and Mostowski said ‘homogeneous set’.) If A is the closure of X under Skolem functions (as we can always arrange), A is said to be an Ehrenfeucht-Mostowski model of T. Ehrenfeucht-Mostowski models have tightly controlled properties. For example they realise few types (see their use in section 6 below). By choosing (X,<) and (X,< ) suﬃciently diﬀerent, we can often ensure that the Ehrenfeucht-Mostowski models constructed over these two ordered sets are not isomorphic; this is the basic idea underlying most of Shelah’s constructionsoflargefamiliesofnonisomorphicmodels(againseesection6). Onecan also construct Ehrenfeucht-Mostowski models of inﬁnitary theories, using various theorems of the Erd˝os-Rado partition calculus in place of Ramsey’s

Theorem. As a byproduct we get a versatile way of building two-cardinal models, i.e. models of ﬁrst-order theories in which some deﬁnable parts have one inﬁnite cardinality and others have another inﬁnite cardinality, as Morley showed in 1965 [140]. (Vaught had obtained two-cardinal results earlier by other methods.) In 1964 Frederick Rowbottom showed that if the set-theoretic universe contains a measurable cardinal (or even an Erd˝os cardinal), then the constructible universe forms an Ehrenfeucht-Mostowski model on a class of ordinals which includes all uncountable cardinals. This result had enormous repercussions in set theory, long before its late publication in [174].

(2) In his 1950 Congress address Tarski [208] showed how to give a precise mathematical deﬁnition of elementary equivalence, using essentially the set-theoretic deﬁnition of satisfaction. This demonstrated that elementary equivalence is a sound notion, but it gave no new information about the notion. It was natural to ask if the same relation could be deﬁned by a completely diﬀerent approach. Any two elementarily equivalent saturated structures of the same cardinality are isomorphic. In 1961 Keisler [104] exploited this to show, with the help of the generalised continuum hypothesis, that two structures are elementarily equivalent if and only if they have isomorphic ultrapowers. Ten years later Shelah [180] proved the same theorem without assuming the generalised continuum hypothesis. Simon Kochen in 1961 [110] gave another characterisation of elementary equivalence, using direct limits of ultrapowers. In spite of their elegance, these characterisations of ≡ had few practical consequences. A much more serviceable answer came from another source. Roland Fra¨ıss´e described a hierarchy of interrelated families of partial isomorphisms between structures [62]. In terms of this hierarchy he gave necessary and suﬃcient conditions for two relational structures to agree in all prenex ﬁrst-order sentences with at most n alternations of quantiﬁer, for each ﬁnite n. SoA ≡ B if A and B agree in this sense for all ﬁnite n. Fra¨ıss´e’s paper was unfortunately hard to read, and his ideas became known through a paper of Ehrenfeucht ([49], 1961) who had come on them independently. Soon afterwards they were rediscovered again by the Kazak mathematician Ta˘ımanov [200]. In Ehrenfeucht’s version, two players play a game to compare two structures A and B. The players alternate; in each step, the ﬁrst player chooses an element of one structure and the second player then chooses an element of the other structure. The second player loses as soon as the elements chosen

from one structure satisfy a quantiﬁer-free formula not satisﬁed by the corresponding elements from the other structure. This is the Ehrenfeucht-Fra¨ıss´ e back-and-forth game on the two structures. For a ﬁrst-order language with ﬁnitely many relation and individual constant symbols and no function symbols, one could show that A and B agree in all sentences of quantiﬁer rank at most k if and only if the second player has a strategy that keeps her alive for at least k steps. Hence A is elementarily equivalent to B if and only if for each ﬁnite k, the second player can guarantee not to lose in the ﬁrst k steps. With this equipment it’s very easy to show that if G,G are elementarily equivalent groups and H,H are elementarily equivalent groups, then the product group G×H is elementarily equivalent to G×H. This is perhaps the best way to view the technique of Feferman and Vaught [58] 1962 for computing the set of sentences true in an arbitrary product of structures. A similar technique worked with ordered sums of structures. The beauty of this idea of Fra¨ıss´e and Ehrenfeucht was that nothing tied it to ﬁrst-order logic. Ehrenfeucht himself [49] used it to prove the equivalence of various ordinal numbers as ordered sets with predicates for + and ·, in a language with a second-order quantiﬁer ranging over ﬁnite sets. Carol Karp [103] adapted it to inﬁnitary logics, and it reappeared in Chang’s construction of Scott sentences [32]. Later (see section 9) it became one of the central tools of computer science logic.

5 Maps between structures

During the period 1930–1950, mathematicians generally had begun to take a closer interest in the maps between structures. This was the period that saw the invention of category theory. The trend naturally made its way into model theory. For example Garrett Birkhoﬀ [21] published his famous characterisation of the classes of models of sets of identities in 1935. Birkhoﬀ’s paper uses a number of straightforward model-theoretic facts about mappings, for example that universally quantiﬁed equations are preserved under taking homomorphic images; in 1951 Marczewski [136] extended this result to all positive ﬁrst-order sentences and asked for a converse. Tarski [210] reported that his own work on formulas preserved in substructures (the SLo´s-Tarski Theorem) was done in 1949–50. This work of SLo´s and Tarski, together with Marczewski’s question, launched a ﬂood of preservation theorems for all

kinds of mapping. Keisler had a rich crop to survey already in 1965 [105]. But theorems of this kind continued to be proved up to the end of the 20th century (e.g. [10] in 1999). Deﬁnability theorems are a close relative of preservation theorems, and they tend to have similar proofs. Instead of maps between structures, they talk about reducts, where one or more symbols are stripped away from the language (as with some forgetful functors). The ﬁrst and most famous deﬁnability theorem of ﬁrst-order model theory was Beth’s Theorem, [20] 1953, which says that if in models of a theory T a certain symbol R is not deﬁnable in terms of the other symbols, then one can ﬁnd two models of T which are identical in everything except the interpretation of the symbol R; in other words, Padoa’s method always works. An unkind comment—though it has some truth—is that in practice one only ever uses the trivial direction of a preservation or deﬁnability theorem. These theorems contribute more to the inner structure of model theory than they do to applications. But during the 1950s the maps between structures came to play a deeper role in model theory, not just as possible topics but as essential tools of the subject. One can trace this development to two model theorists, Abraham Robinson and Roland Fra¨ıss´e. I begin with Robinson.

Robinson’s 1950 Congress address [162] uses embeddings, homomorphisms, elementary embeddings and unions of chains. Elementary embeddings were a new idea. Robinson had no name for them and not much of a general theory. In fact it was Robinson’s style, then and later, to use maps without even having a symbol for them. To ﬁnd an embedding of A in B, he would prove that B is a model of the diagram of A; in Robinson’s sense a diagram was a set of sentences describing a structure, not a display of arrows. It was only in 1957 that Tarski and Vaught [213] deﬁned elementary embeddings and made them generally available. Robinsonhadaparticulargeniusforﬁndingmodel-theoreticstatements— usually about mappings—that are equivalent to signiﬁcant facts in this or that branch of algebra or ﬁeld theory. To measure his style one should compare his treatment of algebraically closed ﬁelds with that of Tarski [207]. Tarski had applied the method of quantiﬁer elimination to the theory T of algebraically closed ﬁelds and established that the quantiﬁer-free formulas form a basic set, and that T has countably many completions, namely one for each characteristic. Robinson made the following observations: (1) If A and B are algebraically closed ﬁelds of the same characteristic, then by downward L¨owenheim-Skolem and compactness one can ﬁnd ﬁelds A, B elementarily equiv

alent to A, B respectively, and both of transcendence degree ω. By Steinitz’ Theorem A and B are isomorphic, and hence A and B are elementarily equivalent. This gives Tarski’s conclusion on completions of T ([162]). (2) We say that a theory S is model-complete if every embedding between models of S is elementary. A suﬃcient condition for model-completeness is that every embedding between models of S preserves universal formulas ([166]). It then follows at once from the Hilbert Nullstellensatz that the theory T of algebraically closed ﬁelds is model-complete ([165]). (3) Suppose that whenever A and B are models of a theory S and X is a nonempty substructure of both A and B, the formulas satisﬁed by elements of X in A are the same as those satisﬁed by the same elements in B; then S has the property of quantiﬁer elimination (i.e. the quantiﬁer-free formulas are a basic set). In the case of our theory T, standard facts about amalgamation of ﬁelds allow us to embed both A and B in an algebraically closed ﬁeld C, forming a commutative diagram over X. By model-completeness the embeddings of A and B in C are elementary, so the criterion for quantiﬁer elimination holds ([168] §4.3). Thus Robinson reached all Tarski’s conclusions about algebraically closed ﬁelds by general model-theoretic principles and standard algebraic facts about ﬁelds. This is typical of Robinson’s reorganisation of the subject. Robinson certainly didn’t conﬁne himself to ﬁnding new proofs of known theorems(thoughhewasneverinhibitedaboutdoingpreciselythis). Among his many contributions were some new examples of model-complete theories. One of these was the theory of diﬀerentially closed ﬁelds of characteristic 0 [165] p. 134 in 1956, as if to predict how useful this theory would be for Hrushovski’s proof of the geometric Mordell-Lang Conjecture some thirtyﬁve years later (section 9 below). Betterknownthanallofthesewasnonstandard analysis, whichappeared almost without warning in 1961 [167]. Robinson used compactness to form an elementary extension R of the ﬁeld R of real numbers (with any further structure attached) containing inﬁnitesimal elements. He noted that if a theorem of real analysis can be written as a ﬁrst-order sentence φ, then to prove φ it suﬃces to use the inﬁnitesimals to show that φ is true in R (a typical example of what Robinson called a transfer argument).

We turn to Fra¨ıss´e. In 1953 he published a short paper [60], with a fuller account a year later [61]. He limited himself to structures with just one relation symbol and no other nonlogical constants (though his arguments are valid for structures with ﬁnitely many relation symbols and individual constants). Taking the ordered set of rational numbers as a paradigm, he

made two important observations. (a) We can characterise those classes of ﬁnite structures which are of the form: all ﬁnite structures embeddable in a given countable structure A. (Following Fra¨ıss´e I shall call these γclasses—it is not a standard name.) (b) A γ-class has the amalgamation property if and only if A can be chosen to be homogeneous, and in this case A is determined up to isomorphism by the γ-class. (A class K has the amalgamation property if for all embeddings e1 : A → B1 and e2 : A → B2 within K there are embeddings f1 : B1 → C and f2 : B2 → C, also within K, such that f0e0 = f1e1. A is homogeneous if every isomorphism between ﬁnite substructures of A extends to an automorphism of A.) Thus Fra¨ıss´e introduced the amalgamation property to model theory (though the name came later, from J´onsson). Fra¨ıss´e’s (a) introduced into model theory a kind of Galois theory of structures: it invited one to think of a structure as built up by a pattern of amalgamated extensions of smaller structures. This idea became important in stability theory. Fra¨ıss´e’s (b) provided a way of building countable structures by assembling a suitable γ-class of ﬁnite structures. His version of the idea was modest, but it was widely used as a source of ω-categorical structures. It was also enough to provide a framework for constructions by Ehud Hrushovski which solved key problems posed by Zilber ([92], [92]). In 1956/7 Bjarni J´onsson, who had reviewed Fra¨ıss´e’s [60], wrote two papers [101], [102] removing the limitation to ﬁnite and countable structures in Fra¨ıss´e’s construction of homogeneous structures. The cost he had to pay was that the generalised continuum hypothesis was needed at some cardinals. Michael Morley realised almost at once (and later published with Vaught who had come to similar conclusions independently, [141] 1962) that, thanks to the Compactness Theorem, J´onsson’s assumptions on the γ-class are veriﬁed if one considers the class of all ‘small’ subsets of models of a completetheory T andreplacesembeddingsbypartialelementarymaps. (Infact Morley and Vaught used a trick from Skolem [192], adding relation symbols so that partial elementary maps become embeddings.) It also emerged that the resulting homogeneous structures were exactly the saturated models of T. Inthe1970stherewassomedebateabouthowbesttohandletheMorleyVaught γ-class. Gerald Sacks [176] proposed one should think of it as a category with partial elementary maps as morphisms. Shelah [185] (Chapter I §1)wentstraighttoaverylargesaturatedmodel C (butweneveraskexactly how large); in his picture the γ-class is simply the class of all small subsets of the domain of C, and the partial elementary maps are the restrictions of

automorphisms of C. Shelah’s view prevailed. The structure C was known as the big model or (following John Baldwin) the monster model. When geometric model theory came on the scene, people noted that by going with Shelah the model-theoretic community had opted for the analogue of Andr´ e Weil’s ‘universal domain’ [221] Chapter IX §1, rather than the more recent category-theoretic language of Grothendieck. The Morley-Vaught theory tells us that under suitable set-theoretic assumptions, every structure has a saturated elementary extension. These set-theoretic assumptions were always a stumbling block, and so weak forms of saturation were devised that served the same purposes without special assumptions. For example every structure has an elementary extension that is special [34]. Every countable structure has a recursively saturated elementary extension [15]. In 1964 Jan Mycielski [146] noticed that Kaplansky’s notion of an algebraically compact abelian group (today more often called a pure-injective abelian group) has a purely model-theoretic characterisation that is a close analogue of saturation. With colleagues in WrocSlaw, Mycielski developed this observation into a theory of atomic compact structures, which was useful on the borderline between model theory and universal algebra. Since atomic compact structures have a large amount of symmetry, they tend to have neat algebraic structural descriptions too; in fact this was the reason for Kaplansky’s interest in them. To some extent the same holds for saturated structures, and even for κ-saturated structures when κ is large enough. For example in 1970 Paul Eklof and Edward Fisher [53] noted that every ω1-saturated abelian group is algebraically compact, and so one can read oﬀ the results of Wanda Szmielew’s quantiﬁer elimination for abelian groups [199] rather easily from Kaplansky’s structure theory. Likewise Ershov [54] used ω1-saturated boolean algebras to recover Tarski’s quantiﬁer elimination results for boolean algebras. Clean methods of this kind quickly became standard practice. Fra¨ıss´e’s contribution was ignored in many accounts of the history. After some careful acknowledgments by Morley and Vaught [141], [61] disappears clean from the record, remembered only by a few people working on ωcategoricity. Maybe Fra¨ıss´e had less of a ﬂair for publicity than Robinson. Another factor may have been that many people at the time regarded the whole line of work from Fra¨ıss´e to Morley and Vaught as trivial. Thus Saunders Mac Lane [128] reports that when Morley ﬁrst brought him the material that led to [141], ... I said, in eﬀect: “Mike, applications of the compactness the

orem are a dime a dozen. Go do something better.” Mac Lane adds that Morley’s Theorem (see section 6 below) was the fruit of this advice. Another factor that may have obscured the role of Fra¨ıss´e and of J´onsson was that model theorists formed the habit of hiding amalgamations. In 1956 Robinson [164] proved a fundamental amalgamation theorem of ﬁrst-order logic. But he never stated it; you have to extract it from Robinson’s proof of a less interesting theorem (the Joint Consistency Theorem). When stability theory returned to Fra¨ıss´e’s view of a structure as built up from amalgamations of extensions of smaller structures, the convention was always to reduce to amalgamations of the form ‘Amalgamate Y ⊃ X and X∪{b}over X’, in line with Keisler’s ‘one element at a time’ approach. Amalgamations of this kind were called extending the type of b over X to Y. In 1983 Shelah [186] restored the amalgamation viewpoint with a vengeance: to construct structures of cardinality ωn from countable pieces, he formed n-dimensional amalgams. But this work of Shelah had little inﬂuence in ﬁrst-order model theory. In any event, some of the best tools of 1960s model theory were a blend of the Robinson line and the Fra¨ıss´e line. For example Robinson’s criterion for a theory T to have quantiﬁer elimination was rewritten as: ‘Given models B and C of T with a common nonempty substructure A, and an enough-saturated elementary extension D of B, there exists an elementary embedding of C into D making the diagram commute.’ To taste, one could require D to be an ultrapower of A. James Ax and Simon Kochen put the new machinery to work in 1965/6 [5], [6], [7] by ﬁnding a complete set of axioms for the ﬁeld of p-adic numbers (uniformly for any prime p) and then showing that this theory has elimination of quantiﬁers. Their method was completely diﬀerent from the method of quantiﬁer elimination, and it seems likely that any proof by that method would have been hopelessly unwieldy. Instead they considered saturated valued ﬁelds of cardinality ω1. One can say a good deal in algebraic terms about the structure of such ﬁelds; Ax and Kochen were able to show that under certain conditions, any two such ﬁelds are isomorphic. They then wrote down these conditions as a ﬁrst-order theory T. Assuming the generalised continuum hypothesis, any two countable models A,B of T have saturated elementary extensions of cardinality ω1, which are isomorphic, so that A and B must be elementarily equivalent. This proves the completeness of T (and hence its decidability since the axioms are eﬀectively enumerable); a similar argument using saturated structures shows that T is model-complete,

and one more push shows that Robinson’s criterion for quantiﬁer elimination is satisﬁed. There are various tricks that one can use to eliminate the generalised continuum hypothesis. This work of Ax and Kochen, together with very similar but independent work of Yuri Ershov, [55], marked the beginning of a long line of research in the model theory of valued ﬁelds. But it hit the headlines because it gave a proof of an ‘almost everywhere’ version of a conjecture of Emil Artin on C2 ﬁelds. Since counterexamples to the full conjecture appeared at about the same time, ‘almost everywhere’ was about as much as one could hope for, short of an explicit list of the exceptions.

Around 1970 category theory was developing fast. Now that model theory took maps between structures seriously, it was reasonable to try to develop a categorical model theory where the maps were the main levers. Michael Makkai and colleagues did some groundwork (e.g. [129]), but the subject never really took oﬀ. Perhaps model theorists enjoy handling elements and dislike morphisms between theories. Nevertheless two papers did show that useful ideas might come from category theory. One was by Daniel Lascar [116] 1982, who had visited Makkai and discussed with him the category of elementary embeddings between models of a complete theory; Lascar’s enquiries threw up several useful ideas, including a notion of Lascar strong type that came to play a role in the study of simple theories. And Edmund Robinson [171] 1986 put categorical logic to good use in a model-theoretic study of the p-adic spectra of commutative rings.

6 Categoricity and classiﬁcation theory

E. V. Huntington [97] called attention to the fact that some theories have only a single model up to isomorphism; he called such theories ‘suﬃcient’. The second-order Peano axioms for number theory (Dedekind [39]) are a familiar example (though not Huntington’s). Oswald Veblen ([219] 1904) proposed the name categorical for theories with this property. As Veblen noted, acategoricaltheoryisnecessarilycompleteinthesensethatitentails, for every sentence φ of the appropriate language, either φ or ¬φ. (Bolzano knew a version of this in 1837: [24] §110.) No ﬁrst-order theory with inﬁnite models can be categorical, by the Upward L¨owenheim-Skolem Theorem. But often in algebra we make do with less: for example if A and B are vector spaces over the same ﬁeld k (and we can express this with ﬁrst-order sentences true in A and B),

then A and B are isomorphic whenever they have the same dimension. In particular if A and B have the same cardinality and it is greater than that of k, then they have the same dimension and so are isomorphic. So it is natural to say (as did Vaught in 1954 [214]) that a theory T is λ-categorical if it has, up to isomorphism, exactly one model of cardinality λ. Then—as Vaught noted, and we saw in section 5 that Robinson [168] had already used a version of the argument—a ﬁrst-order theory which has no ﬁnite models and is λ-categorical for some λ must be complete. In 1959 Lars Svenonius [198] showed that among countable structures, the models of ω-categorical theories are precisely those structures whose automorphism group has ﬁnitely many orbits of n-element sets, for each ﬁnite n. Permutation groups with this property are said to be oligomorphic. Svenonius’ characterisation crossed the boundary between two diﬀerent branches of mathematics, with consequences that we come back to in section 8. Other model theorists (notably CzesSlaw Ryll-Nardzewski [175]) gave purely model-theoretic equivalents of ω-categoricity. In 1955 SLo´s [121] asked: If T is a complete theory in a countable ﬁrstorder language, and T is λ-categorical for some uncountable λ, then is T λ-categorical for every uncountable λ? With hindsight we can see that this was an extraordinarily fortunate question to have asked in 1955, for two main reasons. The ﬁrst was that at just this date the tools for starting to answer the question were becoming available. If T is λ-categorical and A, B are models of T of cardinality λ which are respectively highly saturated and Ehrenfeucht-Mostowski, then A and B are isomorphic and we deduce that models of T of cardinality λ have very few types to realise. This is strong information. Thus SLo´s’s question ‘stimulated quite a bit of the work concerning models of arbitrary complete theories’ (Vaught [216]). Second, SLo´s’s question was unusual in that it called for a description of all the uncountable models of a theory. The answer would involve ﬁnding a structure theorem to explain how any model of the theory is put together. This pointed in a very diﬀerent direction from Tarski’s [210] ‘mutual relations between sentences of formalized theories and mathematical systems in which these sentences hold’. One mark of the change of focus was that expressions like uncountably categorical (i.e. λ-categorical for all uncountable λ) and totally categorical (i.e. λ-categorical for all inﬁnite λ), which originallyappliedtotheories, cametobeusedchieﬂyformodels ofthosetheories. For example Walter Baur wrote in 1975 [16] of ‘ℵ0-categorical modules’. In 1965 Michael Morley answered SLo´s’s question in the aﬃrmative [139]; this is Morley’s Theorem. Amid all the literature of model theory, Morley’s paper stands out for its clarity, its elegance and its richness in original ideas.

Morley’s central innovation was Morley rank, which assigns an ordinal rank to each deﬁnable relation in any model of a theory Tλ-categorical for some uncountable λ. (In Morley’s presentation the rank was assigned to complete types, but later workers generally used the induced rank on formulas or deﬁnable relations.) In an algebraically closed ﬁeld the Morley rank of an algebraic set is equal to its Krull dimension; Morley certainly had some such correlation in mind, as he signalled by giving the name totally transcendental to theories that assign a Morley rank to all deﬁnable relations in their models. Morley conjectured that the Morley rank of any uncountably categorical structure (i.e. the Morley rank of the formula x = x) is always ﬁnite; this was proved soon afterwards by Baldwin [8], and independently by Zilber. Baldwin and Lachlan [9] in 1970 reworked and strengthened Morley’s results. Building on the unpublished dissertation of William Marsh [137], they showed that each model of an uncountably categorical theory carries a deﬁnable strongly minimal set with an abstract dependence relation that deﬁnes a dimension for the model. Once the strongly minimal set is given, the rest of the model is assembled around it in a way that is unique up to isomorphism. They also showed that the number of countable models of such a theory, up to isomorphism, is either 1 or ω. A few young researchers set to work to extend Morley’s result to uncountable ﬁrst-order languages. One of them was Frederick Rowbottom, who in 1964 [173] introduced the name ‘λ-stable’ for theories with few types over sets of λ elements; hence the name stability theory for this general area. In 1969 [178] Saharon Shelah began to publish in stability theory. With his formidable theorem-proving skill he reshaped the subject almost from the start (and many other model theorists ﬂed from the ﬁeld rather than compete with him). By 1971 he had proved the uncountable analogue of Morley’s Theorem [180]. But more important, he had formulated a plan of action. Ehrenfeucht [49] had already noticed that a theory which deﬁnes an inﬁnite linear ordering on n-tuples of elements must have a large number of non-isomorphic models of the same cardinality. Shelah saw this result as marking a division between ‘good’ theories that have few models of the same cardinality, and ‘bad’ theories that have many. Shelah’s strategy was to hunt for possible bad features that a theory might have (like deﬁning an inﬁnite linear ordering), until the list was so comprehensive that a theory without any of these features is pinned down to the point where we can list all of its models in a structure theorem. As Shelah once explained it in conversation, the outcome should be to show that whenever K is the class of all models

of a complete ﬁrst-order theory, ‘if K is good, it is very very good, but if K is bad it is horrid’. Shelah coined the word nonstructure for the horrid case, and he suggested several deﬁnitions of nonstructure [187]. In one deﬁnition, a nonstructure theorem ﬁnds a family of 2λ models of cardinality λ, none of which is elementarily embeddable in any other. In another deﬁnition, a nonstructure theorem ﬁnds two nonisomorphic models of cardinality λ that are indistinguishable by strong inﬁnitary languages. Pursuing this planned dichotomy, Shelah wrote some dozens of papers and one large and famously diﬃcult book ([181] 1978; the second edition in 1990 reports the successful completion of the programme for countable ﬁrst-order theories in 1982). Shelah also wrote a number of papers on analogous dichotomies for inﬁnitary theories or abstract classes of structures (e.g. [185]). His own name for this area of research was classiﬁcation theory. The name applies at two levels: ﬁrst-order theories classify structures, and Shelah’s theory classiﬁes ﬁrst-order theories. Shelah himself often said that his main interests lay on the nonstructure side ([188] p. 154): I was attracted to mathematics by its generality, its ability to give information where apparently total chaos prevails, rather than by its ability to give much concrete and exact information where we a priori know a great deal. But even though he downplayed it himself, in his work on the structure side of the dichotomy he vastly expanded the range of the new tools introduced by Morley. Thus Shelah’s superstable, stable and simple were successive weakenings of Morley’s ‘totally transcendental’, and his forking was a powerful abstract notion of dependence for stable theories. (Shelah didn’t say much about simple theories. But when some examples became important in work of Hrushovski in the early 1990s, Byung-Han Kim [107] showed that forking behaves well in them too.) Shelah’s regular types are a generalisation of strongly minimal sets, in the sense that they carry an abstract dependence relation that gives them a dimension. In a superstable structure, the relations between the regular types determine for example whether we can expand one part of the structure while keeping another part ﬁxed. Shelah also showed that for models of a stable theory, any complete type is in a certain sense ‘deﬁnable’ by ﬁrst-order formulas ([181] 1971, independently proved by Lachlan [111] 1972). He showed that the deﬁnition can always be taken over a canonical base which is a family of imaginary elements of the model (in the sense mentioned in (d) of section 3 above). A special case of his construction is Andr´e Weil’s ([221] p. 68) ﬁeld of deﬁnition

of a variety, except that the ﬁeld of deﬁnition consists of ordinary elements, not imaginary ones. Bruno Poizat explained this in 1985 ([159] §16e) by showing that algebraically closed ﬁelds have elimination of imaginaries, in the sense that their genuine elements can stand in for their imaginary ones. A byproduct of the work of Morley and Shelah was a series of papers determining what structures in various natural classes were categorical, totally transcendental and so forth. The ﬁrst nontrivial paper of this kind was by Joseph Rosenstein, [172] 1969. But certainly the most inﬂuential was a paper of Angus Macintyre in 1972 [124], where he showed that an inﬁnite ﬁeld is totally transcendental if and only if it is algebraically closed. In the aftermath of Macintyre’s paper, Zilber proved (1977 [225]) that any totally transcendental skew ﬁeld is an algebraically closed ﬁeld, and Cherlin and Shelah showed (1979 [37]) that the same is true for superstable skew ﬁelds. Inthecourseofthisandrelatedwork, bothZilberandCherlinindependently noticed that a group deﬁnable in an uncountably categorical structure has many of the typical features of an algebraic group. Cherlin [35] in 1979 conjectured that every simple group of ﬁnite Morley rank is up to isomorphism an algebraic group over an algebraically closed ﬁeld. This became known as Cherlin’s Conjecture; Zilber [225] had conjectured the same thing for the special case of simple groups interpretable in uncountably categorical structures. Stable groups turned out to have an unexpectedly large amount of structure, much of which carried over to modules (which are always stable). Various authors (among them Macintyre, Garavaglia, Baldwin, Saxl, Belegradek) noticed chain conditions that hold in some or all stable groups. In 1985 Poizat [159] created a rich theory of stable groups by generalising ideas from [225] and [37]. Poizat’s framework allows one to rely on intuitions from algebraic geometry in handling stable groups; for example their behaviour is strongly inﬂuenced by their generic elements. At the same time Alexandr Borovik [27] started to bring recent group theory to bear on Cherlin’s Conjecture. Stable groups attracted other workers and remained a lively topic to the end of the century, though Cherlin’s Conjecture is still open.

7 Geometric model theory

Geometric model theory classiﬁes structures in terms of their combinatorial geometries and the groups and ﬁelds that are interpretable in the structures. The roots of this theory go back to work of Lachlan, Cherlin and above all Zilber in stability theory in the 1970s, and for this reason the theory is also

known as geometric stability theory (the title of Pillay’s text [156]). But by the early 1990s it emerged that the same ideas sometimes worked well in structures that were by no means stable. Anabstractdependencerelationgivesrisetoacombinatorialgeometry— in what follows I say just ‘geometry’. In this geometry certain sets of points are closed, i.e. they contain all points dependent on them. Zilber [227] classiﬁed geometries into three classes: (a) trivial or degenerate, where all sets of points are closed; (b) nontrivial locally modular, which are not trivial but if a ﬁnite number of points are ﬁxed (i.e. made dependent on the empty set), then the resulting lattice is modular—for brevity this case is often called modular; (c) the remainder, known brieﬂy as non-modular. Classical examples are: for (a), the dependence relation where an element is dependent only on sets containing it; for (b), linear dependence in a vector space; for (c), algebraic dependence in an algebraically closed ﬁeld. This classiﬁcation made its way into model theory rather indirectly. Zilber was working on a proof that no complete totally categorical theory is ﬁnitely axiomatisable. (His ﬁrst announcement of his proof of this result in 1980 [226] was ﬂawed by a writing-up error which is repaired in [231].) In work on ω-categorical stable theories Lachlan [112] had introduced a combinatorial structure which he called a pseudoplane. A key step in Zilber’s argument was to show that no totally categorical structure contains a deﬁnable pseudoplane. From this he deduced that the geometry of the strongly minimal set must be either trivial or modular, and his main result followed in turn from this. Cherlin, on reading [226] and seeing the error, went to the classiﬁcation of ﬁnite simple groups and proved directly [36] that the strongly minimal set must be either trivial or modular. This result has a purely group-theoretic formulation. In fact several people discovered it independently, and it became known as the Cherlin-Mills-Zilber theorem in honour of three of them. Zilber’s proof, once repaired, reaches the result without the classiﬁcation of ﬁnite simple groups. In the light of Zilber’s work on uncountable categoricity and its extension by Cherlin, Harrington and Lachlan [36], model theorists looked to see what other structures might have modular geometries. One particularly inﬂuential result was proved independently by Hrushovski and Pillay, and published jointly [95]: a group G is modular (i.e. has only modular or trivial geometries) if and only if all deﬁnable subsets of Gn are boolean combinations of cosets of subgroups. We saw that Zilber ﬁrst applied his trichotomy of geometries by showing that in the structures he was considering, the non-modular case never occurred. Zilber now proposed to apply the same trichotomy to another

question, namely his question (mentioned in the previous section) whether every simple group interpretable in an uncountably categorical structure must be an algebraic group over an algebraically closed ﬁeld. Algebraically closed ﬁelds themselves have non-modular geometry; at the 1984 International Congress Zilber [229] conjectured the converse, viz. that any uncountably categorical structure with non-modular geometry must be—up to interpretability both ways—an algebraically closed ﬁeld. This was known as Zilber’s Conjecture. A word about Zilber’s motivation may be in order. Macintyre said in 1988 [127] that ‘Purely logical classiﬁcation[s] give only the most superﬁcial general information’ (and attributed the point to Kreisel). Zilber was convinced that the opposite must be true: if classical mathematics rightly recognises certain structures as ‘good’, then it should be possible to say in purely model-theoretic terms what makes these structures good. In fact Zilber in conversation quoted Macintyre’s paper [124] as an example of how a purely model-theoretic condition (total transcendence) can be a criterion for an algebraic property (algebraic closure). Zilber was also convinced that being a model of an uncountably categorical countable ﬁrst-order theory is an extremely strong property with rich mathematical consequences, among them strong homogeneity and the existence of a deﬁnable dimension. In1988HrushovskirefutedZilber’sConjecture[92]. ButforbothHrushovski and Zilber this meant only that the right condition hadn’t yet been found. Since it seemed to be particularly hard to recover the Zariski topology from purely model-theoretic data, a possible next step was to axiomatise the Zariski topology. This is not straightforward: it has to be done in all ﬁnite dimensions simultaneously, since the closed sets in dimension n don’t determine those in dimension n +1. But Hrushovski described a set of axioms, and Zilber and Hrushovski found that by putting together what they knew, they could prove [96] that Zilber’s Conjecture holds for models of the axioms. In particular, if the geometries are non-modular then the model is isomorphic to the topology of a smooth curve over an algebraically closed ﬁeld, and the ﬁeld is interpretable in the topology. In 1996 Hrushovski [90] published a proof of the geometric Mordell-Lang Conjecture in all characteristics. Key ingredients of his argument were the results on the Zariski topology and on weakly normal groups, and earlier results on the stability of separably closed and diﬀerentiably closed ﬁelds. Hrushovski went on to apply a similar treatment to the Manin-Mumford Conjecture. This case was a little diﬀerent: the structures in question were unstable. But Hrushovski showed that they inherited enough stability from a surrounding algebraically closed ﬁeld; and in any case they were ‘simple’

in Shelah’s classiﬁcation. About the same time, Yakob Peterzil and Sergei Starchenko [155] found that Zilber’s geometric trichotomy gave a strong classiﬁcation of o-minimal ﬁelds. These ﬁelds are a very long way from being stable. By the mid 1990s workers in geometric model theory had built up a large body of expertise. One theme that is worth exploring in this is the role of groups interpretable in a structure. When Baldwin and Lachlan [9] in 1971 had shown that every uncountably categorical structure consists of a strongly minimal set D and other elements attached around it, they found they needed to say something about the way these other elements are attached. Because of categoricity, something in the theory has to prevent the set of attached elements being larger than D. The simplest guess would be that each attached element has to satisfy an algebraic formula (i.e. one satisﬁed by only ﬁnitely many elements) withparametersin D. BaldwinandLachlanﬁnishedtheirpaperwithacomplicated example to show that this need not hold. Later Baldwin realised that an easy example was already to hand: a direct sum G of countably many cyclic groups of order p2 for a prime p. The socle (the set of elements of order at most p) is strongly minimal, in fact a vector space over the pelement ﬁeld. An element a of order p2 is described by saying what pa is; but if b is any element of the socle then some automorphism of G ﬁxes the socle pointwise and takes a to a + b. In fact the orbit of a over the socle is parametrised by elements of the socle. This parametrisation keeps the orbit from having cardinality greater than that of the socle. Zilber [231] realised that this was a common pattern in uncountably categorical structures. Each such structure is a ﬁnite tower; at the bottom is a strongly minimal set, and as we go up the tower, the orbit of an element over the preceding level in the tower is always parametrised by some group interpretable in that preceding level. He called these groups binding groups. There are some cohomological constraints, which allowed Ahlbrandt and Ziegler

[2] to begin cataloguing the possibilities. Later Zilber ([228] Lemma 3.3) called attention to the combinatorial conﬁguration

which occurs in modular strongly minimal sets. (The blobs are points of the geometry. All points are pairwise independent. A line between three points means they form a dependent set.) He showed how to construct a group from the conﬁguration; but since this was in the middle of an argument by reductio ad absurdum and quite strong assumptions were in force, it was less than the deﬁnitive result. Hrushovski [89] looked closer and showed, using Zilber’s conﬁguration, that every modular regular type has an inﬁnite group interpretable in it (in a generalised sense). Soon after this, Hrushovski [94] and Laskowski [118] used this result of Hrushovski to show that the power of groups to bind structures together could be used to settle unsolved problems of stability theory. This made it clear that henceforth the interpretability of groups in a structure would have to be considered one of the tools of model theory. A number of researchers went to work to discover and characterise the groups interpretable in various structures of model-theoretic interest.

8 Model theory within mathematics

Model-theoretic writers before 1950 rarely said what they thought about the relation between their work and the rest of mathematics. The natural assumption was that they were exploring properties of notions in the foundations of mathematics: axiom, independence, consistency, model, truth. Tarski ([211] p. 152) saw his 1933 truth deﬁnition as a contribution to one of ‘the classical problems of philosophy’. Listening to the contributions of Robinson and Tarski to the 1950 International Congress, we begin to hear a new tune. Both speakers set out to convince their audience that contemporary symbolic logic can produce useful tools—though by no means omnipotent ones—for the development of actual mathematics, more particularly for the development of algebra and, it would appear, algebraic geometry (Robinson [162] p. 694) and that model theory has applications which may be of general interest to mathematicians and especially to algebraists; in some of these applications the notions of [model theory] itself are not involved at all’ (Tarski

[208] p. 717). Over the next few decades, claims of this kind became commonplace, especially in applications for research grants.

There were some high points in the search for applications ‘in which the notions of model theory itself are not involved’. Nonstandard analysis was one. Another was the work of Ax, Kochen and Ershov on Artin’s Conjecture on C2 ﬁelds (see section 5 above). In 1984 Jan Denef

[41] used Macintyre’s quantiﬁer elimination for p-adic ﬁelds in order to prove a special case of a conjecture of Serre on Poincar´e series; then with van den Dries, Denef proved the full conjecture

[42] in 1988. Murthy and Swan in 1976

[145] gave an application of a criterion of Eklof [51] 1975 for a fact of geometry to be independent of the choice of universal domain. Shelah [182] applied stability theory to show that in characteristic 0 each diﬀerential ﬁeld has a unique prime diﬀerential closure. And although it took some time to ﬁlter out of Russia, Mal’tsev’s early work on local theorems in group theory was a signiﬁcant example. In all these cases a device from model theory—usually involving compactness—was used to prove a result deﬁnitely belonging to some other area of mathematics. There was a revealing episode in the late 1960s. James Ax [4] gave a brief and neat model-theoretic proof of the ‘somewhat unexpected fact that an injective morphism of an algebraic variety into itself is surjective’ (Armand Borel’s description). Borel [27] and Shimura promptly found proofs not using model theory; then M. Raynaud pointed out (in a footnote to Borel’s paper) that Grothendieck had essentially proved the result already in 1967 by sheaf methods. The outcome seemed to be that the model-theoretic proof was elegant but unnecessary, and Ax’s result owed more to his skill as a mathematician than it did to his model-theoretic approach. It seems not to have inspired other geometers or number theorists to learn model theory. There is a similar story to tell about Robinson’s use of nonstandard analysis to solve the problem whether square roots of compact operators on Hilbert spaces have invariant subspaces ([19], 1966). In a telephone interview with Dauben ([38] p. 327), Chang rightly said ‘Major credit must go to Robinson’; but he said ‘to Robinson’, not ‘to nonstandard analysis’. One-oﬀ successes of this kind were never a likely way to win converts to model theory from outside. Sometimes model theorists claimed to have better model-theoretic proofs of known results from other branches of mathematics; this was even less calculated to win friends and inﬂuence people. In fact the idea of ‘applying model theory to actual mathematics’ was already growing rusty by 1970. By that date most model theorists regarded model theory itself as a part of actual mathematics. And in any case the natural ebb and ﬂow of mathematical research throws up much subtler relationships than ‘applying area X in area Y’. For example two areas may overlap; questions or methods of common interest form a weak overlap, and

a stronger overlap is where the same researchers place themselves in both ﬁelds. Certainly model theory had for a time a strong overlap of this kind with set theory. Tarski noted already in 1950 that ([208] p. 705) ‘set-theoretical constructions and methods play an essential part in the development of the general theory of arithmetical classes’. During the 1960s there was a good deal of interplay between model theory and set theory. Set theory gave conditions for the existence of various kinds of model, and one of the ﬁrst applications of Ronald Jensen’s ﬁne structure theory was to a twocardinal question in model theory [99]. In return model theory gave set theoryindiscerniblesandultrapowers. Gaifman, RowbottomandSilverwere three of the people who could be found on both sides of the divide. Though few would have predicted it, this proved to be only a temporary alliance. In the early 1970s something of a revulsion against combinatorial set theory began to express itself among model theorists. Within ﬁrst-order model theory the developments of sections 3 to 5 above seemed to have run their natural course, and the future lay in applications or in inﬁnitary analogues. The model theory of inﬁnitary logic seemed in danger of lapsing into axiomatic or descriptive set theory. (One could quote the work of Shelah, Eklof and Mekler [52] on almost free abelian groups as an example of the former, and Vaught [217] and Barwise [11] to illustrate the latter.) The one exception to prove the rule was Shelah, who brought in theorems of F˝odor and Solovay as essential tools of nonstructure theory, and invented proper forcing for applications in model theory. Shelah’s vigorous use of combinatorial set theory, beautiful mathematics though it was, unfortunately did little to encourage younger researchers to move into the area opened up by Morley, until Lascar and Poizat [117] published their elegant and set-theory-free introduction to stability theory in 1979. The tide might yet turn between model theory and set theory. During the 1990s several authors pointed to links between descriptive set theory and current ﬁrst-order model theory, not least among them Vaught’s Conjecture [17]. The model theory of sets had a kind of younger brother in the model theory of arithmetic, which provided a door between model theory and proof theory. From Skolem to the end of the 20th century, many model theorists examined models of (ﬁrst-order) Peano arithmetic and other fragments of ﬁrst-order arithmetic. Often these were test cases for general theorems of model theory. But John Shepherdson, in an inﬂuential paper of a more distinctive kind [190] 1965, proposed using models of fragments of arithmetic as a way of studying the proof-theoretic strength of these fragments. This

led to a line of papers in which the model theory and the proof theory of arithmetic were closely intertwined. Most famously, Jeﬀ Paris and Leo Harrington [150] in 1977 used a model-theoretic construction to show that the consistency of Peano arithmetic follows from Peano arithmetic together with a result intermediate between the ﬁnite and inﬁnite Ramsey theorems (thus providing the ﬁrst ‘natural’ arithmetical theorem undecidable from the Peano axioms). The frontier with universal algebra was also friendly. Model theory and universal algebra had a common interest in at least the classes axiomatised by universal Horn theories (for example Mal’tsev [133] 1973 or more recently Hart et al. [77] 1994), and they both talked about embeddings, homomorphisms and direct products in a very general setting. But there was not much traﬃc across this frontier. The equation “model theory = universal algebra + logic” from Chang and Keisler [33] never meant much in practice. There was also a brief and small-scale liaison with graph theory. Tony Gardiner [66], when in 1976 he classiﬁed the ﬁnite homogeneous simple graphs, had no idea that he was classifying those ﬁnite simple graphs whose complete ﬁrst-order theory has the quantiﬁer elimination property. When model theorists noticed it, they saw scope for a range of similar classiﬁcations. One of the ﬁrst was the classiﬁcation of countable homogeneous graphs by Lachlan and Woodrow [113] in 1980 (a paper that set new standards in beauty for model-theoretic illustrations). But no longterm links were established; Gardiner’s paper had already almost exhausted the matters of common interest. Although set theory, universal algebra and graph theory are unambiguously parts of ‘actual mathematics’, none of them lies close to the central areas of algebra and algebraic geometry that Robinson and Tarski hoped to engage with. Before I turn to these areas, I note what happened to nonstandard analysis almost as soon as Robinson introduced it. The subject had (it seems) very little to give back to model theory, and almost at once it formed its own largely separate community. Turning to algebra, one deﬁnite success—though on a small scale—was the study of countable models of ω-categorical theories. As we noted in section 6, this is simply the study of oligomorphic permutation groups. But the equivalence was not cashed in until Dugald Macpherson, a combinatorialist writing his DPhil under Peter Cameron, approached Lachlan in 1981. Zilber’s work on ω-categorical groups (see section 6 above) became known soon after, and this brought the group theorist Peter Neumann and his DPhil student David Evans onto the scene. This was enough to form the seed of a research community in the area of overlap. The resulting harvest included

[29], [83], two works that are equally in model theory and in group theory. At the end of the century the pace of work had died down, but there was still unﬁnished business, for example on covers and on smoothly approximable structures. Relations with ﬁeld theory were harder to pin down, and I have to rely on personal impressions. After the work of Ax, Kochen and Ershov reported in section 5, not many ﬁeld theorists persevered in the area. Michael Fried and Moshe Jarden were an exception, as witness their book

[65] 1986. But on the model theory side, a number of people built up a great deal of expertise in one or other area of ﬁeld theory and function theory. (Names that come to mind are Chatzidakis, Delon, van den Dries, Gardener, Haskell, Macintyre, Marker, Scanlon, Speissegger, Wilkie, among others.) This group were capable of taking on classical open problems and solving them using model-theoretic tools; a typical example was the solution by van den Dries, Macintyre and Marker of an old problem of Hardy, [47]. Members of this group were highly respected by ‘classical’ mathematicians who knew little about model theory. But the communities never really merged. One noticed that at conferences where members of this group met, there were rarely more than one or two classical geometers or function theorists—and not for lack of invitations. In1996[91]HrushovskipublishedtheﬁrstproofofthegeometricMordellLang Conjecture in all characteristics. The language of his proof was modeltheoretic, [ASK McQUILLAN FOR A COMMENT?] At the end of the century it was still unclear whether Hrushovski’s work had opened up an area that model theorists and algebraic geometers would be able to cultivate together. There were some signs that members of the two communities wanted a closer contact. A number of joint papers by model theorists and geometers appeared. Hrushovski himself acted as an interpreter between the communities, for example i

[94].9 Other languages, other structures

In 1885 C. S. Peirce [153], fresh from inventing quantiﬁers, mentioned that the universal and the existential quantiﬁer are not the only examples. He gave the example of the quantiﬁer ‘For two-thirds of all x’. Unfortunately nobody picked up Peirce’s idea, until in 1957 Mostowski [144] called attention to the quantiﬁers ‘For at least ℵα x’. Mostowski’s paper was timely, becauseitwasusefultohaveinthe1960savarietyofextensionsofﬁrst-order logic for testing out new constructions.

In 1969

[120] Per Lindstr¨om published another timely paper, in which he gave model-theoretic necessary and suﬃcient conditions for a logic to have the same expressive power as ﬁrst-order logic. His result suggested that it might be possible to ﬁt the various logics studied during the previous decade into some higher organisation of logics, within a generalised (or abstract) model theory. Alas, the facts weren’t there to support such a theory. The 1970s saw some valiant eﬀorts in this direction, and by the mid 1980s a large amount was known about many diﬀerent logics extending ﬁrst-order logic (see Barwise and Feferman [13]). But the most quotable outcome of any generality was that very few logics apart from ﬁrst-order logic satisfy the Craig interpolation lemma. Themathematicallogicianswithincomputerscienceshruggedtheirshoulders and asked what is the interest of a logic in which it’s impossible to express everyday notions like connectedness, even on ﬁnite structures. Thus for example Gurevich [75] 1984: Thequestionariseshowgoodisﬁrst-orderlogicinhandlingﬁnite structures. It was not designed to deal exclusively with ﬁnite structures. ... One would like to enrich ﬁrst-order logic so that the enriched logic ﬁts better the case of ﬁnite structures. One solution was ﬁrst-order logic with a ﬁxed-point operator added, as in Chandra and Harel [31] 1982 and Blass, Gurevich and Kozen [23] 1985. The model theory of this logic and its relatives were studied mostly by computer scientists, but this seems to be purely an accident of history; these languages would have been good to have available in the 1960s. For other various other reasons, logicians studied logics with only a ﬁnite number of variables. (Barwise [12] 1977 was the ﬁrst of several people who independently described these logics and their associated pebble games.) Other logicians studied logics of a modal kind, with or without ﬁxed-point operators. EhrenfeuchtFra¨ıss´e back-and-forth games adapt smoothly to these contexts, and there are computer science applications. During the 1990s the group of Jouko V¨a¨an¨anen in Helsinki took a particular interest in game-theoretic and combinatorial aspects of non-classical logics both large and small (e.g. [98] 1990, [80] 1996). One of the early variants of ﬁrst-order logic was still showing signs of life in the 1990s. In 1961 Henkin [82] described an extension of ﬁrst-order logic where the dependencies between the quantiﬁers are not controlled by the order of the quantiﬁers, for example as in For all x there is y such that for all z there is w depending only on z such that R(x,y,z,w).

Sometimes, as in this example, the dependencies between the quantiﬁers could be written as a branching diagram; in which case one spoke of branching quantiﬁers. Henkin proposed a semantics in terms of Skolem functions, and Hintikka [87] later revised it into the form of games of imperfect information. There was some debate whether these logics had useful applications in computer science (e.g. Blass and Gurevich [22] 1986) or in linguistics. But in themselves they had a problematic model theory, because the semantics determines only when a sentence is true in a structure, and not whether an assignment of elements satisﬁes a formula in a structure. Thus one could ask about classes deﬁned by axioms in such a logic, but not about relations deﬁnable by a formula in a structure. Work to clarify this continued to the end of the century.

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[171] Edmund P. Robinson, The p-adic spectrum, Journal of Pure and Applied Algebra 40 (1986) 281–296. [172] Joseph G. Rosenstein,ℵ0-categoricity of linear orderings, Fundamenta Mathematicae 64 (1969) 1–5.

[173] Frederick Rowbottom, The SLo´s conjecture for uncountable theories, Notices of American Mathematical Society 11 (1964) 248.

[174] Frederick Rowbottom, Some strong axioms of inﬁnity incompatible with the axiom of constructibility, Annals of Mathematical Logic 3 (1971) 1–44.

[175] CzesSlaw Ryll-Nardzewski, On the categoricity in powerℵ0, Bulletin de l’Acad´emie Polonaise des Sciences S´erie Scientiﬁque et Math´ematique (CHECK) 7 (1959) 545–548.

[176] Gerald E. Sacks, Saturated Model Theory, W. A. Benjamin, Reading Mass 1972. [177] Dana S. Scott, Measurable cardinals and constructive sets, Bulletin de l’Acad´emie Polonaise des Sciences S´erie Scientiﬁque, Math´ematique, Astronomique, Physique CHECK 9 (1961) 521–524. [178] Saharon Shelah, Stable theories, Israel Journal of Mathematics 7 (1969) 187–202.

[179] Saharon Shelah, The number of non-isomorphic models of an unstable ﬁrst-order theory, Israel Journal of Mathematics 9 (1971) 473–487. [180] Saharon Shelah, Every two elementarily equivalent models have isomorphic ultrapowers, Israel Journal of Mathematics 10 (1971) 224–233.

[181] Saharon Shelah, Stability, the f.c.p., and superstability; model theoreticpropertiesofformulasinﬁrstordertheory, Annals of Mathematical Logic 3 (1971) 271–362. [182] SaharonShelah, Uniquenessandcharacterizationofprimemodelsover sets for totally transcendental ﬁrst-order theories, Journal of Symbolic Logic 37 (1972) 107–113.

[183] Saharon Shelah, Categoricity of uncountable theories, in Proceedings of Tarski Symposium, ed. L. Henkin et al., American Mathematical Society, Providence RI 1974, pp. 187–203. [184] Saharon Shelah, Inﬁnite abelian groups, Whitehead problem and some constructions, Israel Journal of Mathematics 18 (1974) 243–256. [185] Saharon Shelah, Classiﬁcation Theory and the Number of NonIsomorphic Models, North-Holland, Amsterdam 1978. [186] Saharon Shelah, Classiﬁcation theory for nonelementary classes I, The number of uncountable models of ψ ∈ Lω1ω, Israel Journal of Mathematics 46 (1983) 212–240 and 241–273.

[187] Saharon Shelah, Classiﬁcation of ﬁrst order theories which have a structure theorem, Bulletin of American Mathematical Society 12 (1985) 227–232.

[188] Saharon Shelah, Taxonomy of universal and other classes, in Proceedings of International Congress of Mathematicians, 1986, ed. Andrew M. Gleason, American Mathematical Society pp. 154–162. [189] Saharon Shelah, Classiﬁcation theory for non-elementary classes II, Abstract Elementary Classes, in Classiﬁcation Theory, ed. J. T. Baldwin, Lecture Notes in Mathematics 1292, Springer, Berlin 1987, pp. 419–497.

[190] John C. Shepherdson, Non-standard models for fragments of number theory, in The theory of models, Proceedings of 1963 International Symposium at Berkeley, ed. J. W. Addison et al., North-Holland, Amsterdam 1965, pp. 342–358.

[191] Thoralf Skolem, Untersuchungen ¨uber die Axiome des Klassenkalk¨uls und ¨uber Produktations- und Summationsprobleme, welche gewisse Klassen von Aussagen betreﬀen, Videnskapsselskapets Skrifter, I. Matem.-naturv. klasse no. 3 (1919).

[192] Thoralf Skolem, Logisch-kombinatorische Untersuchungen ¨uber die Erf¨ullbarkeit und Beweisbarkeit mathematischen S¨atze nebst einem Theoreme ¨uber dichte Mengen, Videnskapsselskapets Skrifter, I. Matem.-naturv. klasse I no. 4 (1920) 1–36.

[193] Thoralf Skolem, Einige Bemerkungen zur axiomatischen Begr¨undung der Mengenlehre, in Proceedings of 5th Scandinavian Mathematical Congress, Helsinki 1922, pp. 217–232. [194] ThoralfSkolem, ¨UbereinigeSatzfunktioneninderArithmetik, Skrifter utgitt av Det Norske Videnskaps-Akademi i Oslo I, no. 7 (1931) 1–28; reprinted in [197] pp. 281–306. [195] Thoralf Skolem, ¨Uber die Nichtcharakterisierbarkeit der Zahlenreihe mittels endlich oder abz¨ahlbar unendlich vieler Aussagen mit ausschliesslich Zahlenvariablen, Fundamenta Mathematicae 23 (1934) 150– 161. [196] Thoralf Skolem, A critical remark on foundational research, Kongelige Norske Videnskabsselskabs Forhandlinger, Trondheim 28, no. 20 (1955) 100–105; reprinted in [197] pp. 581–586.

[197] Thoralf Skolem, Selected Works in Logic, ed. Jens Erik Fenstad, Universitetsforlaget, Oslo 1970.

[198] Lars Svenonius,ℵ0-categoricity in ﬁrst-order predicate calculus, Theoria 25 (1959) 82–94.

[199] Wanda Szmielew, Elementary properties of Abelian groups, Fundamenta Mathematicae 41 (1955) 203–271. [200] A. D. Ta˘ımanov, Characteristics of axiomatisable classes of models (Russian), Algebra i Logika 1 (4) (1962) 5–31.

[201] Alfred Tarski, Sur les ensembles d´eﬁnissables de nombres r´eels I, Fundamenta Mathematicae 17 (1931) 210–239. [202] Alfred Tarski, revised version translated as The Concept of Truth in Formalized Languages, [211] pp. 152–278.

[203] Alfred Tarski, Grundz¨uge des Systemenkalk´uls, Fundamenta Mathematicae 25 (1935) 503–526 and 26 (1936) 283–301; tr. as Foundations of the calculus of systems, in [211] pp. 342–383.

[204] Alfred Tarski, O poj¸eciu wynikania logicznego, Przegl¸ad Filozoﬁczny 39 (1936) 58–68; tr. as On the concept of logical consequence, in [211] pp. 409–420.

[205] Alfred Tarski, Introduction to Logic and to the Methodology of Deductive Sciences, Oxford University Press, New York 1994 (Polish original 1936).

[206] Alfred Tarski, The Completeness of Elementary Algebra and Geometry, Institut Blaise Pascal, Paris 1967 (a reprint of a work scheduled to appear in 1940 but prevented by the war).

[207] Alfred Tarski, Arithmetical classes and types of algebraically closed and real-closed ﬁelds, Bulletin of American Mathematical Society 55 (1949) 64. [208] Alfred Tarski, Some notions and methods on the borderline of algebra and metamathematics, in Proceedings of International Congress of Mathematicians, Cambridge Mass. 1950, vol. 1, pp. 705–720.

[209] Alfred Tarski, A decision method for elementary algebra and geometry (prepared for publication with the assistance of J.C.C. McKinsey), 2nd edition, University of California Press, Berkeley 1951.

[210] Alfred Tarski, Contributions to the theory of models I, Indagationes Mathematicae 16 (1954) 572–581. [211] Alfred Tarski, Logic, Semantics, Metamathematics, trans. J. H. Woodger, second edition ed. and introduced by John Corcoran, Hackett, Indianapolis 1983.

[212] Alfred Tarski, A. Mostowski and R. M. Robinson, Undecidable Theories, North-Holland, Amsterdam 1953. [213] Alfred Tarski and Robert L. Vaught, Arithmetical extensions of relational systems, Compositio Mathematica 13 (1957) 81–102.

[214] Robert L. Vaught, Applications of the L¨owenheim-Skolem-Tarski theorem to problems of completeness and decidability, Indagationes Mathematicae 16 (1954) 467–472. [215] Robert L. Vaught, Denumerable models of complete theories, in Inﬁnitistic methods, Proc. Symp. Foundations of Math., Warsaw 1959, Pa´nstwowe Wydawnictwo Naukowe, Warsaw 1961, pp. 303–321.

[216] Robert L. Vaught, Models of complete theories, Bulletin of American Mathematical Society 69 (1963) 299–313.

[217] Robert L. Vaught, Descriptive set theory in Lω1ω, inCambridge Summer School in Mathematical Logic, ed. A. R. D. Mathias and H. Rogers, Lecture Notes in Mathematics 337, Springer, Berlin 1973, pp. 574–598.

[218] Robert L. Vaught, Alfred Tarski’s work in model theory, Journal of Symbolic Logic 51 (1986) 869–882.

[219] Oswald Veblen, A system of axioms for geometry, Transactions of American Mathematical Society 5 (1904) 343–384.

[220] Hao Wang, A survey of Skolem’s work in logic, in [197], pp. 17–52.

[221] Andr´e Weil, Foundations of Algebraic Geometry, American Mathematical Society Colloquium Publications vol. 29, American Mathematical Society, New York 1946.

[222] Hermann Weyl, ¨Uber die Deﬁnitionen der mathematischen Grundbegriﬀe, Mathematisch-naturwissenschaftliche Bl¨atter 7 (1910) 93–95 and 109–113. [223] Alex J. Wilkie, Model completeness results for expansions of the ordered ﬁeld of real numbers by restricted Pfaﬃan functions and the exponential function, Journal of American Mathematical Society 9 (1996) 1051–1094.

[224] Martin Ziegler, Algebraisch abgeschlossene Gruppen, in Word Problems II, The Oxford Book, ed. S.I. Adian et al., North-Holland, Amsterdam 1980, pp. 449–576. 321–329. [225] B. I. Zil’ber, Groups and rings whose theory is categorical (Russian), Fundamenta Mathematicae 9 (1977) 173–188; tr. in American Mathematical Society Translations 149 (1991) 1–16. [226] Boris I. Zil’ber, Strongly minimal countably categorical theories (Russian), Sibirskii Matematicheskii Zhurnal 21 (1980) 98–112.

[227] Boris I. Zil’ber, Totally transcendental structures and combinatorial geometries (Russian), Doklady Akademii Nauk SSSR 259 (1981) 5, pp. 1039–1041.

[228] Boris I. Zil’ber, Strongly minimal countably categorical theories (Russian), Sibirskii Matematicheskii Zhurnal Part II, 25 No. 3 (1984) 71–88; Part III, 25 No. 4 (1984) 63–77.

[229] Boris I. Zil’ber, The structure of models of uncountably categorical theories, in Proceedings of the International Congress of Mathematicians, Warszawa 1983, vol. I, PWN, Warsaw and North-Holland, Amsterdam 1984, pp. 359–368.

[230] Boris I. Zil’ber, Finite homogeneous geometries, in Proceedings of Sixth Easter Conference in Model Theory, ed. Bernd Dahn et al., Humboldt-Univ. 1988, pp. 186–208. [231] Boris I. Zil’ber, Uncountably categorical theories, trans. D. Louvish, Translations of Mathematical Monographs 117, American Mathematical Society, Providence RI 1993.

Several model theory texts give more detailed historical information about particular theorems; for example Chang and Keisler [33], Hodges [88], Pillay [156]. There are surveys on Skolem by Hao Wang [220] and on Tarski by Vaught [218]. Sadly we still lack an authoritative survey of Abraham Robinson’s contributions to model theory.

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