Some Applications of Set Theory to Model Theory

Abstract

We investigate set-theoretic dividing lines in model theory. In particular, we are interested in Keisler's order and Borel complexity.

Keisler's order is a pre-order on complete countable theories $T$, measuring the saturation of ultrapowers of models of $T$. In Chapter~\ref{SurveyChapter}, we present a self-contained survey on Keisler's order. In Chapter~\ref{KeislerNew}, we uniformize and sharpen several ultrafilter constructions of Malliaris and Shelah. We also investigate the model-theoretic properties detected by Keisler's order among the simple unstable theories.

Borel complexity is a pre-order on sentences of $\mathcal{L}_{\omega_1 \omega}$ measuring the complexity of countable models. In Chapter~\ref{ChapterURL}, we describe joint work with Richard Rast and Chris Laskowski on this order. In particular, we connect the Borel complexity of $\Phi \in \mathcal{L}_{\omega_1 \omega}$ with the number of potential canonical Scott sentences of $\Phi$. In Chapter~\ref{ChapterSB}, we introduce the notion of thickness; when $\Phi$ has class-many potential canonical Scott sentences, thickness is a measure of how quickly this class grows in size. In Chapter~\ref{ChapterTFAG}, we describe joint work with Saharon Shelah on the Borel complexity of torsion-free abelian groups.