Indestructibility properties of Ramsey and Ramsey-like cardinals

Abstract

We develop new techniques, for use in indestructibility arguments, of lifting embeddings on transitive set models of ZFC− which lack closure and of lifting iterations of such embeddings to a forcing extension. We use these techniques to establish basic indestructibility results for Ramsey and the Ramsey-like cardinals – α-iterable, strongly Ramsey, and super Ramsey cardinals – introduced in [11]. We show that Ramsey, α-iterable, strongly Ramsey, and super Ramsey cardinals κ are indestructible by small forcing, the canonical forcing of the GCH, and the forcing to add a fast function on κ. We also show that if κ is one of these large cardinals, then there is a forcing extension in which the large cardinal property of κ becomes indestructible by Add(κ,θ) for any cardinal θ.The following are consequences of the indestructibility results. If κ is Ramsey, α-iterable, strongly Ramsey, or super Ramsey, then there is a forcing extension preserving this in which the GCH fails at κ. If κ is one of these large cardinals, then there is a forcing extension preserving the large cardinal property of κ in which κ is not even weakly compact in HOD. If κ is Ramsey, then there is a forcing extension in which κ remains virtually Ramsey, but is no longer Ramsey (this answers positively a question posed in [11]).