The Keisler-Shelah isomorphism theorem and the continuum hypothesis II

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 $\mathbf{m}=⟨({\mathbb{M}}_n^1,{\mathbb{M}}_n^2:n<\omega ⟩$ 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 $\prod _{\mathcal{D}}{\mathbb{M}}_n^1$ and $\prod _{\mathcal{D}}{\mathbb{M}}_n^2$ are elementarily equivalent, then they are isomorphic.

arXiv commentThis is publication 1223 of the second author