A Note on Indestructibility and Strong Compactness

Arthur W. Apter¹

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 56 (2008) no. 3, pp. 191-197

Résumé

If $\kappa \lambda$ are such that $\kappa$ is both supercompact and indestructible under $\kappa$-directed closed forcing which is also $(\kappa^+, \infty)$-distributive and $\lambda$ is $2^\lambda$ supercompact, then by a result of Apter and Hamkins [J. Symbolic Logic 67 (2002)], $\{\delta \kappa \mid \delta$ is $\delta^+$ strongly compact yet $\delta$ is not $\delta^+$ supercompact$\}$ must be unbounded in $\kappa$. We show that the large cardinal hypothesis on $\lambda$ is necessary by constructing a model containing a supercompact cardinal $\kappa$ in which no cardinal $\delta > \kappa$ is $2^\delta = \delta^+$ supercompact, $\kappa$'s supercompactness is indestructible under $\kappa$-directed closed forcing which is also $(\kappa^+, \infty)$-distributive, and for every measurable cardinal $\delta$, $\delta$ is $\delta^+$ strongly compact if{f} $\delta$ is $\delta^+$ supercompact.

Détail

DOI : 10.4064/ba56-3-1

Keywords: kappa lambda kappa supercompact indestructible under kappa directed closed forcing which kappa infty distributive lambda lambda supercompact result apter hamkins symbolic logic delta kappa mid delta delta strongly compact yet delta delta supercompact unbounded kappa large cardinal hypothesis lambda necessary constructing model containing supercompact cardinal kappa which cardinal delta kappa delta delta supercompact kappas supercompactness indestructible under kappa directed closed forcing which kappa infty distributive every measurable cardinal delta delta delta strongly compact delta delta supercompact