The Keisler-Shelah isomorphism theorem and the continuum hypothesis

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 $CH$.

arXiv commentThis is publication 1215 of second author