Submission date: 10 August 2012.

Abstract:

We give the first (ZFC) dividing line in Keisler's order among the unstable theories, specifically among the simple unstable theories. That is, for any infinite cardinal λ for which there is μ < λ \leq 2^μ, we construct a regular ultrafilter D on λ such that (i) for any model M of a stable theory or of the random graph, M^λ/D is λ^+-saturated but (ii) if Th(N) is not simple or not low then N^λ/D is not λ^+-saturated. The non-saturation result relies on the notion of flexible ultrafilters. To prove the saturation result we develop a property of a class of simple theories, called Qr1, generalizing the fact that whenever B is a set of parameters in some sufficiently saturated model of the random graph, |B| = λ and μ < λ \leq 2^μ, then there is a set A with |A| = μ such that any non-algebraic p in S(B) is finitely realized in A. In addition to giving information about simple unstable theories, our proof reframes the problem of saturation of ultrapowers in several key ways. We give a new characterization of good filters in terms of “excellence”, a measure of the accuracy of the quotient Boolean algebra. We introduce and develop the notion of {moral} ultrafilters on Boolean algebras. We prove a so-called “separation of variables” result which shows how the problem of constructing ultrafilters to have a precise degree of saturation may be profitably separated into a more set-theoretic stage, building an excellent filter, followed by a more model-theoretic stage: building moral ultrafilters on the quotient Boolean algebra, a process which highlights the complexity of certain patterns, arising from first-order formulas, in certain Boolean algebras.

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Full text arXiv 1208.2140: pdf, ps.