Keisler’s Order Has Infinitely Many Classes

We prove, in ZFC, that there is an infinite strictly descending chain of classes of theories in Keisler’s order. Thus Keisler’s order is infinite and not a well order. Moreover, this chain occurs within the simple unstable theories, considered model-theoretically tame. Keisler’s order is a central notion of the model theory of the 60s and 70s which compares first-order theories (and implicitly ultrafilters) according to saturation of ultrapowers. Prior to this paper, it was long thought to have finitely many classes, linearly ordered. The model-theoretic complexity we find is witnessed by a very natural class of theories, the n-free k-hypergraphs studied by Hrushovski. This complexity reflects the difficulty of amalgamation and appears orthogonal to forking. A significant challenge to our understanding of unstable theories in general, and simple theories in particular, has been the apparent intractability of the problem of Keisler’s order. Determining the structure of this order is a large-scale classification program in model theory. Its structure on the stable theories was known, and recent progress on the unstable case has had surprising applications, described in [24], [25], [27]. The order was long thought to be finite, with perhaps four classes, whose identities were suggested in 1978 (see Problem 0.3 below). In the present paper, we leverage the ZFC theorems of [26] to prove, nearly fifty years after Keisler introduced the order, that Keisler’s order has infinitely many classes, and moreover is not a well order. The nature of this infinite hierarchy suggests that the order may encode much more model-theoretic information than was generally thought. There are a number of recent accounts of Keisler’s order, as in the introduction to [26]. We give here a brief sketch to put our main theorem in context. Keisler’s order asks about the saturation properties of certain limits of sequences of models, the regular ultrapowers. If D is an ultrafilter on the infinite set I, let us call D regular if whenever M is a model in a countable language, whether or not the ultrapower M /D is |I|-saturated depends only on the theory of M . (The theorem giving this equivalent definition is due to Keisler [12]. In fact, consistently, all ultrafilters are regular [2].) Given a theory T and a regular ultrafilter D, we may therefore say that “D saturates T” if indeed M /D is |I|-saturated for some, equivalently every, model of T . Keisler proposed the study of the pre-order on complete, countable theories given by: Definition 0.1 (Keisler 1967). T1 E T2 iff any regular ultrafilter D which saturates T2 must also saturate T1. Date: Thursday 20th October, 2016. 2010 Mathematics Subject Classification. 03C20, 03C45 ; 03E05, 05C65, 06E10. Thanks: Malliaris was partially supported by a Sloan research fellowship, by NSF grant DMS1300634, and by a research membership at MSRI funded through NSF 0932078 000 (Spring 2014). Shelah was partially supported by European Research Council grant 338821. This is paper 1050 in Shelah’s list.