Mixed Levels of Indestructibility

Arthur W. Apter ¹

¹ Department of Mathematics Baruch College of CUNY New York, NY 10010, U.S.A. and The CUNY Graduate Center, Mathematics 365 Fifth Avenue New York, NY 10016, U.S.A.

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) no. 2, pp. 113-122

Résumé

Starting from a supercompact cardinal $\kappa $, we force and construct a model in which $\kappa $ is both the least strongly compact and least supercompact cardinal and $\kappa $ exhibits mixed levels of indestructibility. Specifically, $\kappa $'s strong compactness, but not its supercompactness, is indestructible under any $\kappa $-directed closed forcing which also adds a Cohen subset of $\kappa $. On the other hand, in this model, $\kappa $'s supercompactness is indestructible under any $\kappa $-directed closed forcing which does not add a Cohen subset of $\kappa $.

Détail

DOI : 10.4064/ba63-2-2

Keywords: starting supercompact cardinal kappa force construct model which kappa least strongly compact least supercompact cardinal kappa exhibits mixed levels indestructibility specifically kappa strong compactness its supercompactness indestructible under kappa directed closed forcing which adds cohen subset kappa other model kappa supercompactness indestructible under kappa directed closed forcing which does cohen subset kappa