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 $\lcf(\aleph_0, \de)$ of $\aleph_0$ modulo $\de$, 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, detected by non-low theories. Assuming $κ> \aleph_0$ is measurable, we construct a regular ultrafilter on $λ\geq 2^κ$ which is flexible (thus: ok) but not good, and which moreover has large $\lcf(\aleph_0)$ but does not even saturate models of the random graph. 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. 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 for any $n < ω$, assuming the existence of $n$ measurable cardinals below $λ$, there is a regular ultrafilter $D$ on $λ$ such that any $D$-ultrapower of a model of linear order will have $n$ alternations of cuts, as defined below. Moreover, $D$ will $λ^+$-saturate all stable theories but will not $(2^κ)^+$-saturate any unstable theory, where $κ$ is the smallest measurable cardinal used in the construction.