Existence of optimal ultrafilters and the fundamental complexity of simple theories

Abstract

In the first edition of Classification Theory, the second author characterized the stable theories in terms of saturation of ultrapowers. Prior to this theorem, stability had already been defined in terms of counting types, and the unstable formula theorem was known. A contribution of the ultrapower characterization was that it involved sorting out the global theory, and introducing nonforking, seminal for the development of stability theory. Prior to the present paper, there had been no such ultrapower characterization of an unstable class. In the present paper, we first establish the existence of so-called optimal ultrafilters on (suitable) Boolean algebras, which are to simple theories as Keisler's good ultrafilters [15] are to all (first-order) theories. Then, assuming a supercompact cardinal, we characterize the simple theories in terms of saturation of ultrapowers. To do so, we lay the groundwork for analyzing the global structure of simple theories, in ZFC, via complexity of certain amalgamation patterns. This brings into focus a fundamental complexity in simple unstable theories having no real analogue in stability.