The Keisler-Shelah isomorphism theorem and the continuum hypothesis

TL;DR

This paper explores the relationship between the Keisler-Shelah isomorphism theorem and the continuum hypothesis, showing that certain ultraproduct isomorphisms imply CH and discussing their consistency without CH.

Contribution

It establishes a connection between ultraproduct isomorphisms of elementary structures and the truth of the continuum hypothesis, providing new insights into model theory and set theory.

Findings

Ultraproduct isomorphisms imply CH under certain conditions.

Consistency results for Keisler and Shelah theorems without CH.

Characterization of when ultraproducts reflect set-theoretic assumptions.

Abstract

We show that if for any two elementary equivalent structures $\mathbf{M}, \mathbf{N}$ of size at most continuum in a countable language, $\mathbf{M}^{\omega}/ \mathcal{U} \simeq \mathbf{N}^\omega / \mathcal{U}$ for some ultrafilter $\mathcal{U}$ on $\omega,$ then $CH$ holds. We also provide some consistency results about Keisler and Shelah isomorphism theorems in the absence of $CH$.