A Note on Indestructibility and Strong Compactness

Volume 56 / 2008

Arthur W. Apter Bulletin Polish Acad. Sci. Math. 56 (2008), 191-197 MSC: 03E35, 03E55. DOI: 10.4064/ba56-3-1


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.


  • Arthur W. ApterDepartment of Mathematics
    Baruch College of CUNY
    New York, NY 10010, U.S.A.
    The CUNY Graduate Center, Mathematics
    365 Fifth Avenue
    New York, NY 10016, U.S.A.

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