Indestructibility of generically strong cardinals

Brent Cody ¹   ; Sean Cox ¹

¹ Virginia Commonwealth University Department of Mathematics and Applied Mathematics 1015 Floyd Avenue P.O. Box 842014 Richmond, VA 23284, U.S.A.

Fundamenta Mathematicae, Tome 232 (2016) no. 2, pp. 131-149

Résumé

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 $\omega _1$.

Détail

DOI : 10.4064/fm232-2-3

Keywords: foreman proved duality theorem which gives algebraic characterization certain ideal quotients generic extensions application proved generic supercompactness nbsp omega preserved proper forcing generalize portions foremans duality theorem context generic extender embeddings ideal extenders introduced claverie application prove nbsp omega generically strong remains after adding number cohen subsets nbsp omega however many other nbsp omega closed posets col omega omega destroy generic strongness omega generalizes results gitik shelah nbsp about indestructibility strong cardinals generically strong context prove similar theorems successor cardinals larger omega