A dividing line within simple unstable theories

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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≤2μ, we construct a regular ultrafilter D on λ so that (i) for any model M of a stable theory or of the random graph, Mλ/D is λ+-saturated but (ii) if Th(N) is not simple or not low then Nλ/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≤2μ, then there is a set A with |A|=μ so that any nonalgebraic p∈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 so-called 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. © 2013 Elsevier Inc.




Malliaris, M., & Shelah, S. (2013). A dividing line within simple unstable theories. Advances in Mathematics, 249, 250–288. https://doi.org/10.1016/j.aim.2013.08.027

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