Exact saturation in pseudo-elementary classes for simple and stable theories

TL;DR

This paper investigates PC-exact saturation in stable and simple theories, revealing its role in characterizing stability cardinals, supersimplicity, and supershortness, with implications for set-theoretic properties of models.

Contribution

It provides new characterizations of stability, supersimplicity, and supershortness via PC-exact saturation, including for singular and countable cofinality cardinals.

Findings

PC-exact saturation characterizes stability cardinals of size at least continuum.

Simple unstable theories have PC-exact saturation at certain singular cardinals.

Supersimplicity is characterized by PC-exact saturation at singular cardinals of countable cofinality.

Abstract

We study PC-exact saturation for stable and simple theories. Among other results, we show that PC-exact saturation characterizes the stability cardinals of size at least continuum of a countable stable theory and, additionally, that simple unstable theories have PC-exact saturation at singular cardinals, satisfying mild set-theoretic hypotheses, which had previously been open even for the random graph. We characterize supersimplicity of countable theories in terms of having PC-exact saturation at singular cardinals of countable cofinality. We also consider the local analogue of PC-exact saturation, showing that local PC-exact saturation for singular cardinals of countable cofinality characterizes supershort theories.