Abstract

Using an idea developed in joint work with Shelah, we show how to redefine Laver's notion of forcing making a supercompact cardinal $\kappa$ indestructible under $\kappa$-directed closed forcing to give a new proof of the Kimchi-Magidor Theorem in which every compact cardinal in the universe satisfies certain indestructibility properties. Specifically, we show that if K is the class of supercompact cardinals in the ground model, then it is possible to force and construct a generic extension in which the only strongly compact cardinals are the elements of K or their measurable limit points, every $\kappa \in K$ is a supercompact cardinal indestructible under $\kappa$-directed closed forcing, and every $\kappa$ a measurable limit point of K is a strongly compact cardinal indestructible under $\kappa$-directed closed forcing not changing $\wp$. We then derive as a corollary a model for the existence of a strongly compact cardinal $\kappa$ which is not $\kappa^+$ supercompact but which is indestructible under $\kappa$-directed closed forcing not changing $\wp and remains non-$\kappa^+$ supercompact after such a forcing has been done.