Indestructibility of generically strong cardinals
Volume 232 / 2016
Abstract
Foreman (2013) proved a Duality Theorem which gives an algebraic characterization of certain ideal quotients in generic extensions. As an application he proved that generic supercompactness of $\omega _1$ is preserved by any proper forcing. We generalize portions of Foreman's Duality Theorem to the context of generic extender embeddings and ideal extenders (as introduced by Claverie (2010)). As an application we prove that if $\omega _1$ is generically strong, then it remains so after adding any number of Cohen subsets of $\omega _1$; however many other $\omega _1$-closed posets—such as $ {\rm Col}(\omega _1, \omega _2)$—can destroy the generic strongness of $\omega _1$. This generalizes some results of Gitik–Shelah (1989) about indestructibility of strong cardinals to the generically strong context. We also prove similar theorems for successor cardinals larger than $\omega _1$.
