Constructing regular ultrafilters from a model-theoretic point of view

Abstract:

This paper contributes to the set-theoretic side of understanding Keisler’s order. We consider properties of ultrafilters which affect saturation of unstable theories: the lower cofinality $\mathrm {lcf}(\aleph _0, \mathcal {D})$ of $\aleph _0$ modulo $\mathcal {D}$, 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 and 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 $\lambda \geq 2^\kappa$ which is flexible 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. Since flexible is precisely OK, this also shows that (c) from a set-theoretic point of view, consistently, OK need not imply good, addressing a problem from Dow (1985). Second, under no additional assumptions, we prove that there is a loss of saturation in regular ultrapowers of unstable theories, and also give a new proof that there is a loss of saturation in ultrapowers of non-simple theories. More precisely, for $\mathcal {D}$ regular on $\kappa$ and $M$ a model of an unstable theory, $M^\kappa /\mathcal {D}$ is not $(2^\kappa )^+$-saturated; and for $M$ a model of a non-simple theory and $\lambda = \lambda ^{<\lambda }$, $M^\lambda /\mathcal {D}$ is not $\lambda ^{++}$-saturated. In the third part of the paper, 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 $\mathcal {D}$ 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 $\mathcal {D}$ 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.