The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

TL;DR

This paper explores the relationship between the Keisler-Shelah isomorphism theorem and the continuum hypothesis, demonstrating consistency results where ultraproducts of models are isomorphic under certain conditions even when CH fails.

Contribution

It shows that it is consistent with set theory that the continuum hypothesis fails and ultraproducts of models satisfying specific properties are isomorphic.

Findings

Ultraproducts of models are isomorphic under certain conditions.

Consistency of the failure of CH with ultraproduct isomorphism results.

Extension of previous work on the Keisler-Shelah theorem.

Abstract

We continue the investigation started in [Sh:1215] about the relation between the Keilser-Shelah isomorphism theorem and the continuum hypothesis. In particular, we show it is consistent that the continuum hypothesis fails and for any given sequence $\mathbf m=\langle (\mathbb{M}^{1}_n, \mathbb{M}^{2}_n: n < \omega \rangle$ of models of size at most $\aleph_1$ in a countable language, if the sequence satisfies a mild extra property, then for every non-principal ultrafilter $\mathcal D$ on $\omega$, if the ultraproducts $\prod\limits_{\mathcal D} \mathbb{M}^{1}_n$ and $\prod\limits_{\mathcal D} \mathbb{M}^{2}_n$ are elementarily equivalent, then they are isomorphic.