Indestructibility of generically strong cardinals

摘要

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 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).