Vaught's Conjecture for o-Minimal Theories

Introduction. The notion of o-minimality was formulated by Pillay and Steinhorn [PS] building on work of van den Dries [D]. Roughly speaking, a theory T is ominimal if whenever M is a model of T then M is linearly ordered and every definable subset of the universe of M consists of finitely many intervals and points. The theory of real closed fields is an example of an o-minimal theory. We examine the structure of the countable models for T, T an arbitrary ominimal theory (in a countable language). We completely characterize these models, provided that T does not have 2@' countable models. This proviso (viz. that T has fewer than 2@' countable models) is in the tradition of classification theory: given a cardinal a, if T has the maximum possible number of models of size a, i.e. 2a, then no structure theorem is expected (cf. [Shi]). O-minimality is introduced in ?1. ?1 also contains conventions and definitions, including the definitions of cut and noncut. Cuts and noncuts constitute the nonisolated types over a set. In ?2 we study a notion of independence for sets of nonisolated types and the corresponding notion of dimension. In ?3 we define what it means for a nonisolated type to be simple. Such types generalize the so-called "components" in Pillay and Steinhorn's analysis of w)categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2' countable models. ?4 is devoted to proving that if T has fewer than 2@' countable models, then the isomorphism type of a countable model of T is determined by the order types of its realizations of the nonisolated types in S1(0). Our proof depends on claims to the effect that if T does not have 2' countable models then whenever M is a model of T and X and Y are "pieces" of M which can be described using 0-definable points (i.e. X and Y are sets of realizations in M of types over 0) then (for all n E co) there is at most one M-definable n-ary function which maps X to Y. This analysis extends Pillay and Steinhorn's study of orthogonal components [PS]. In ?5 we use our analysis of the countable models of an o-minimal theory to prove a strong form of Vaught's conjecture for these theories.