Ultrafilter extensions do not preserve elementary equivalence

Denis I. Saveliev, Saharon Shelah

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 $β(\mathcal M_1)…