3 A ug 1 99 8 Universal Indestructibility

From a suitable large cardinal hypothesis, we provide a model with a supercompact cardinal in which universal indestructibility holds: every supercompact and partially supercompact cardinal κ is fully indestructible by <κ-directed closed forcing. Such a state of affairs is impossible with two supercompact cardinals or even with a cardinal which is supercompact beyond a measurable cardinal. Laver’s intriguing preparation [Lav78] makes any supercompact cardinal κ indestructible by <κ-directed closed forcing. In his model, however, cardinals which are only partially supercompact are not generally indestructible; indeed, almost all of them are highly destructible. But this needn’t be so. We aim in this paper to provide a model of a supercompact cardinal with universal indestructibility, one in which every supercompact and partially supercompact cardinal γ is fully indestructible by <γ-directed closed forcing. Main Theorem. If there is a high-jump cardinal, then there is a transitive model with a supercompact cardinal in which universal indestructibility holds. The high jump cardinals, defined below, have a consistency strength above a supercompact cardinal and below an almost huge cardinal. Modified versions of the Main Theorem will provide a model of a strongly compact cardinal in which universal indestructibility holds for strong compactness and a model of a strong cardinal in which universal indestructibility holds for strongness. Let us begin by proving that the Laver preparation itself does not achieve universal indestructibility. Our research has been supported in part by PSC-CUNY grants and by a Collaborative Incentive Grant from the CUNY Research Foundation.