Constructing regular ultrafilters from a model-theoretic point of view

by Malliaris and Shelah. [MiSh:996]
Transactions American Math Soc, 2015
This paper, the first of two, contributes to the set-theoretic side of understanding Keisler's order. We deal with properties of ultrafilters which affect saturation of unstable theories: the lower cofinality {mathrm {lcf}}(aleph_0, {D}) of aleph_0 modulo {D}, saturation of the minimum unstable theory (the random graph), flexibility, goodness, goodness for equality, and realization of symmetric cuts. We work in ZFC except when noted, as several constructions appeal to complete ultrafilters thus assume a measurable cardinal. The main results are as follows. First, we investigate the strength of flexibility, known to be detected by non-low theories. Assuming kappa > aleph_0 is measurable, we construct a regular ultrafilter on lambda >= 2^kappa which is flexible (thus: ok) but not good, and which moreover has large {mathrm {lcf}} (aleph_0) but does not even saturate models of the random graph. This implies (a) that flexibility alone cannot characterize saturation of any theory, however (b) by separating flexibility from goodness, we remove a main obstacle to proving non-low does not imply maximal, and (c) from a set-theoretic point of view, consistently, ok need not imply good, answering a question from Dow 1985. Under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for {D} regular on kappa and M a model of an unstable theory, M^kappa / {D} is not (2^kappa)^+-saturated; and for M a model of a non-simple theory and lambda = lambda^{< lambda}, M^lambda / {D} is not lambda^{++}-saturated. Finally, we investigate realization and omission of symmetric cuts, significant both because of the maximality of the strict order property in Keisler's order, and by recent work of the authors on SOP_2 . We prove that if {D} is a kappa- complete ultrafilter on kappa, any ultrapower of a sufficiently saturated model of linear order will have no (kappa, kappa)-cuts, and that if {D} is also normal, it will have a (kappa^+, kappa^+)-cut. We apply this to prove that for any n < omega, assuming the existence of n measurable cardinals below lambda, there is a regular ultrafilter D on lambda such that any D-ultrapower of a model of linear order will have n alternations of cuts, as defined below. Moreover, D will lambda^+-saturate all stable theories but will not (2^{kappa})^+-saturate any unstable theory, where kappa is the smallest measurable cardinal used in the construction.

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