fea-pillay.qxp 10/16/00 9:03 AM Page 1373 Model Theory Anand Pillay odel theory is a branch of mathemat- pathology. This is of course true in many ways. In ical logic. It has been considered as a fact, possibly the deepest results of twentieth- subject in its own right since the 1950s. century logic, such as the Gödel incompleteness I will try to convey something of the theorems, the independence of the continuum model-theoretic view of the world of hypothesis from the standard axioms of set theory, Mmathematical objects. The use of the word “model” and the negative solution to Hilbert’s Tenth in model theory is somewhat different from (and Problem, have this character. However, from al- even opposed to) usage in ordinary language. In most the beginning of modern logic there have the latter, a model is a copy or ideal representation been related, but in a sense opposite, trends, often of some real object or phenomena. A model in the within the developing area of model theory. In the sense of model theory, on the other hand, is 1920s Tarski gave his decision procedure for supposed to be the real thing. It is a mathematical elementary Euclidean geometry. A fundamental structure, and that which it is a model of (a set of result, the “compactness theorem” for first-order axioms, say) plays the role of the idealization. logic, was noticed by Gödel, Mal’cev, and Tarski in Abraham Robinson was fond of the expression the 1930s and was, for example, used by Mal’cev “metamathematics” as a description of mathemat- to obtain local theorems in group theory. Abraham ical logic. His dream, which he realized in many ways, Robinson in the 1950s introduced differentially was to show how metamathematical arguments and closed fields and discovered nonstandard analysis. considerations can produce new mathematical The eve of the modern era saw James Ax’s decid- results. Logic began as the delineation and classifi- ability for the theory of finite fields and Ax-Kochen’s cation of (logically) valid forms of reasoning. One asymptotic solution to a problem of E. Artin. The might not expect this science of tautology to have modern era has seen, often for unexpected reasons much to say about the world. However, for reasons and sometimes because of the internal development people do not fully understand, it does have some- of the subject, that model theory is in a position thing to say about the mathematical world. to discern amazing patterns and analogies in tame Logic is, in the popular imagination, often mathematics (those areas of mathematics not sub- associated with undecidability, paradoxes, and ject to Gödel undecidability and incompleteness) and in the process to obtain new results. This is Anand Pillay is Swanlund Professor of Mathematics at the what I want to discuss. University of Illinois, Urbana-Champaign. His e-mail I will have to introduce some technical mater- address is [email protected]. He was partially ial, outlining the basic notions and objects of model supported by NSF Grant DMS 00-70179. theory as we see them now. In the process I will This article was influenced by a series of talks given refer back to and explain something of the older by the author at the Southwest Center for Arithmetic results mentioned above, but my aim is rather to Algebraic Geometry in May 2000, and he thanks the number theorists at the University of Arizona, Tucson, for describe current work and prospects for the future. their hospitality. He thanks also Enrique Casanovas and I will be limiting my description and analysis to a Ward Henson for their helpful comments on an earlier few “main trends” in model theory. As such I will draft of this article, as well as the editors for their either not touch on or give scant attention to many encouragement and hard work in bringing it to fruition. interesting fields of research, such as finite model DECEMBER 2000 NOTICES OF THE AMS 1373 fea-pillay.qxp 10/16/00 9:03 AM Page 1374 theory (definability in finite structures, which is re- We want to be able to distinguish clearly between lated to both database theory and computational labels and the objects to which they refer. complexity), universal algebra, nonstandard analy- Given a structure M, say, let us introduce labels sis and the model theory of infinite-dimensional or names for the distinguished functions and sets. spaces, infinitary logic, generalized quantifiers, This choice of labels gives rise to a vocabulary or “abstract” model theory, and models of arithmetic language for the structure M. In fact, the usual way and set theory. of introducing structures is to start with a vocabu- For further reading on model theory in the spirit lary L = (fi,Rj )i,j consisting of function labels and of this article, I recommend starting with the set labels of specified arities (i.e., the integers n volume [4]. A classic text on model theory is [2]. An mentioned above) and then to define an L-structure elegant and very accessible text on o-minimality is M to be a set X equipped with distinguished M M [3]. For more on the connections with diophan- functions fi and sets Rj corresponding to these tine-geometric issues, I recommend [1] and [5]. labels. One often says, for example, that f M is the interpretation of the label f in the structure M. Structures and Definability Different structures can have the same vocabulary. The key notion of model theory, which I think For example, suppose our language L consists cannot be avoided even though there are many of a single binary function label f. We can form an attempts in popular expositions to get around it, L-structure M whose underlying set is the integers Z M is the notion of “truth in a structure”. A structure such that the interpretation f of f is M here is simply a set X, say, equipped with a addition of integers. Another L-structure N is Z distinguished family of functions from Xn to X obtained by again taking as the underlying set but defining f M to be multiplication of integers. (various n) and a distinguished family of subsets 0 of Xn (various n). Here Xn is the Cartesian For another example, suppose the language L consists of a single set label R of arity 2. We can product X X, n times. We shall assume 0 that the diagonal {(x, x):x ∈ X} is among the form an L -structure P whose underlying structure Z P distinguished sets even if it is not explicitly men- is the integers such that the interpretation R of R is the relation set for <, i.e., the subset of tioned. There is not much to say here except that (x, y) ∈ Z2 with x<y. whenever mathematicians focus on a certain object As a matter of convention, = is always the label X (or even a category), they are typically interested for the equality relation on a structure (the diag- not in all subsets of X, X X, etc., but certain onal in X X). ones. For example, if X is an algebraic variety iden- In any case, from the labels for the distinguished tified with its points in some algebraically closed functions and sets in a structure M, together with field, the algebraic geometer will be interested in ∧ symbols for the usual logical operations “and” ( ), the algebraic subvarieties of X, X X, etc., among “or” (∨), “implies” (→), “not” (¬), “there exists” (∃), which of course is the diagonal. So the important “for all” (∀), as well as an infinite supply of thing in looking at a structure is not only the “variables” (x,y,...), we can build up first-order underlying set X but also the family of functions expressions that can be interpreted as making and sets with which it is equipped. The integers statements about the structure M and/or about considered as an additive group (i.e., with addition elements of that structure. The first-order aspect the only distinguished function) is a very different is that the variables range over elements of the structure from the integers considered as a ring. underlying set X (rather than subsets). It should A slightly more general and mathematically quite also be emphasized that a first-order formula is a natural notion is that of a many-sorted structure. In finite string of symbols. the place of a single underlying set or universe X, we What about truth? Suppose f is a label for a unary have an indexed family (Xi)i of universes. The function. Consider the first-order expression : distinguished relations and functions live on or go ∀x∀y(f (x)=f (y) → x = y). between various Cartesian products of the Xi. Every- thing we say below, including the notion of truth In words this expresses that whenever f (x)=f (y), of first-order formulas in a structure and the then x = y. The expression is true or false in M compactness theorem, adapts unproblematically to according to whether or not the function f M from the many-sorted setting. X to X is one-one. This is an example of a first- Nevertheless, let us work with a single underlying order sentence. On the other hand, the expression set X. If fi and Aj are the distinguished functions (x): and sets for a structure M , we write M as ∃y(f (y)=x) (X,fi,Ai)i,j. The additive group of reals becomes (R, +, 0) in this notation.