A Note on Indestructibility and Strong Compactness
Tom 56 / 2008
Streszczenie
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.
