Using Ultrapowers to Compare Continuous Structures

TL;DR

This paper explores a pre-ordering concept in continuous model theory, inspired by earlier work on discrete theories, to classify theories based on ultrapower saturation properties.

Contribution

It extends the ultrapower pre-ordering framework from classical to continuous model theory, providing new tools for theory classification.

Findings

Established an analogous pre-ordering in continuous model theory.

Demonstrated the richness of this pre-ordering for classifying theories.

Connected the classification to ultrapower saturation properties.

Abstract

In 1967 the author introduced a pre-ordering of all first order complete theories where T is lower than U if it is easier for an ultrapower of a model of T than an ultrapower of a model of U to be saturated. In a long series of recent papers, Malliaris and Shelah showed that this pre-ordering is very rich and gives a useful way of classifying simple theories. In this paper we investigate the analogous pre-ordering in continuous model theory.