Vaught's conjecture for o-minimal theories

Abstract

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 o-minimal 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 o-minimal theory (in a countable language). We completely characterize these models, provided that T does not have 2ω countable models. This proviso (viz. that T has fewer than 2ω countable models) is in the tradition of classification theory: given a cardinal α, if T has the maximum possible number of models of size α, i.e. 2α, then no structure theorem is expected (cf. [Sh1]).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 ω-categorical o-minimal theories [PS]. We show that if there is a nonisolated type which is not simple then T has 2ω countable models.