The Keisler–Shelah isomorphism theorem and the continuum hypothesis
Volume 260 / 2023
Fundamenta Mathematicae 260 (2023), 59-66
MSC: Primary 03C20; Secondary 03E35.
DOI: 10.4064/fm198-5-2022
Published online: 8 September 2022
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 related to Keisler and Shelah isomorphism theorems in the absence of CH.
