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2009.58: O-Minimal Structures

2009.58: A.J. Wilkie (2007) O-Minimal Structures. Séminaire BOURBAKI, 60 (985). pp. 1-11. ISSN 0036-8075

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Abstract

The notion of an o-minimal expansion of the ordered field of real numbers was invented by L van den Dries [vdD1] as a framework for investigating the model theory of the real exponential function exp : R ! R : x ! ex, and thereby settle an old problem of Tarski. More on this later, but for the moment it is best motivated as being a candidate for Grothendieck’s idea of “tame topology” as expounded in his Esquisse d’un Programme [Gr]. It seems to me that such a candidate should satisfy (at least) the following criteria. (A) It should be a framework that is flexible enough to carry out many geometrical and topological constructions on real functions and on subsets of real euclidean spaces. (B) But at the same time it should have built in restrictions so that we are a priori guaranteed that pathological phenomena can never arise. In particular, there should be a meaningful notion of dimension for all sets under consideration and any that can be constructed from these by use of the operations allowed under (A). (C) One must be able to prove finiteness theorems that are uniform over fibred collections. None of the standard restrictions on functions that arise in elementary real analysis satisfy both (A) and (B). For example, there exists a continuous function G : (0, 1) ! (0, 1)2 which is surjective, thereby destroying any hope of a dimension theory for a framework that admits all continuous functions. Restricting to the smooth (i.e. C1) 985–02 environment fares no better. For every closed subset of any euclidean space, in particular the subset graph(G) of R3, is the set of zeros of some smooth function. So by the use of a few simple constructions that we would certainly wish to allow under (A), we soon arrive at dimension-destroying phenomena. The same is even true (though this is harder to prove) if we start from just those smooth functions that are everywhere real analytic (i.e. equal the sum of their Taylor series on a neighbourhood of every point), although, as we shall see, this class of functions is locally well-behaved and as such can serve as a model for the three criteria above. Rather than enumerate analytic conditions on sets and functions sufficient to guarantee the criteria (A), (B) and (C) however, we shall give one succinct axiom, the o-minimality axiom, which implies them. Of course, this is a rather open-ended (and currently flourishing) project because of the large number of questions that one can ask under (C). One must also provide concrete examples of collections of sets and functions that satisfy the axiom and this too is an active area of research. In this talk I shall survey both aspects of the theory. Our formulation of the o-minimality axiom makes use of definability theory from mathematical logic. We begin with a collection F of real valued functions of real variables (not necessarily all of the same number of arguments). We consider the ordered field structure on R augmented by the functions in F. This gives us a first-order structure (or model ) RF := hR;+, ·,−,<,Fi, and we denote the corresponding firstorder logical language by L(F). We then call the structure RF o-minimal if whenever (x) is an L(F)-formula (with parameters) then the subset of R defined by (x) is a finite union of open intervals and points (i.e. it is the union of finitely many connected sets). I shall elucidate what is meant by an L(F)-formula and by the subset of R (and, more generally, of Rn) defined by such a formula in the next two sections. However, I should emphasize at this stage that such a formula not only defines a subset , denoted (RF), of Rn, but also a subset (R) of Rn where R is any ordered ring augmented by a collection of functions, F say, such that F and F are in correspondence via a bijection that preserves the number of places (arity) of the functions. One can, and should, define the notion o-minimality for such structures hR;Fi and it was at (rather more than) this level of generality that the true foundations of the subject were laid by Pillay and Steinhorn in [P-S], shortly after van den Dries’ work on the real field. Indeed, it turned out that the solution to Tarski’s problem on the real exponential function (the case F = {exp} in the above notation) relied heavily on the Pillay-Steinhorn theory of o-minimality for structures based on ordered fields other than the reals. This having been said, I shall concentrate in this lecture on the real case, alluding only occasionally to the more general situation, and leave the reader to adapt the definitions and theorems to the setting of o-minimal expansions of arbitrary ordered fields.

Item Type:Article
Subjects:MSC 2000 > 03 Mathematical logic and foundations
MSC 2000 > 11 Number theory
MIMS number:2009.58
Deposited By:Ms Lucy van Russelt
Deposited On:03 October 2009

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