Indestructibility Results for Ramsey Cardinals

We will prove basic indestructibility results for Ramsey cardinals. Ramsey cardinals can be characterized by the existence of certain nontrivial elementary embeddings–Ramsey embeddings–whose domains are certain transitive sets of size \(\kappa\). However, the standard lifting techniques do not quite apply: Ramsey embeddings have domains and targets that need not be closed (not even under countable sequences), and so the usual diagonalization method to build generic filters does not apply. Moreover, to verify that the lifted embedding witnesses that \(\kappa\) is Ramsey, one has to show (among other things) that the ultrafilter generated by the lifted embedding is countably complete. We will present a new diagonalization criterion for models without closure, and we will also present sufficient conditions so that the ultrafilter generated by the lift is countably complete. As a result, we will show that Ramsey cardinals are indestructible by a variety of forcing notions.

This is joint work with Victoria Gitman.