Ultrafilter extensions do not preserve elementary equivalence

TL;DR

The paper demonstrates that ultrafilter extensions of models can fail to preserve elementary equivalence, even when the original models are elementarily embedded.

Contribution

It provides a counterexample showing ultrafilter extensions do not necessarily preserve elementary equivalence between models.

Findings

Ultrafilter extensions of models may not be elementarily equivalent.

Elementary embedding does not imply ultrafilter extension equivalence.

Counterexamples exist for preservation of elementary equivalence in ultrafilter extensions.

Abstract

We show that there exist models $\mathcal M_1$ and $\mathcal M_2$ such that $\mathcal M_1$ elementarily embeds into $\mathcal M_2$ but their ultrafilter extensions $\beta(\mathcal M_1)$ and $\beta(\mathcal M_2)$ are not elementarily equivalent.