Countable models of nonmultidimensional ℵ0-stable theories

Abstract

In this paper T will always be a countable ℵ0-stable theory, and in this introduction a model of T will mean a countable model.One of the main notions we introduce is that of almost homogeneous model: we say that a model M of T is almost homogeneous if for all ā and finite sequences of elements in M, if the strong type of ā is the same as the strong type of (i.e. for all equivalence relations E, definable over the empty set and with a finite number of equivalence classes, ā and are in the same equivalence class), then there is an automorphism of M taking ā to . Although this is a weaker notion than homogeneity, these models have strong properties, and it can be seen easily that the Scott formula of any almost homogeneous model is in L1. In fact, Pillay [Pi.] has shown that almost homogeneous models are characterized by the set of types they realize.The motivation of this research is to distinguish two classes of ℵ0-Stable theories:(1) theories such that all models are almost homogeneous;(2) theories with 2ℵ0 nonalmost homogeneous models.The example of theories with Skolem functions [L. 1] (almost homogeneous is then equivalent to homogeneous) seems to indicate a link between these properties and the notion of multidimensionality, and that nonmultidimensional theories are in the first case.