The Lottery Preparation

The lottery preparation, a new general kind of Laver preparation, works uniformly with supercompact cardinals, strongly compact cardinals, strong cardinals, measurable cardinals, or what have you. And like the Laver preparation, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal κ, for example, becomes fully indestructible by <κ-directed closed forcing; a strong cardinal κ becomes indestructible by ≤κ-strategically closed forcing; and a strongly compact cardinal κ becomes indestructible by, among others, the forcing to add a Cohen subset to κ, the forcing to shoot a club C ⊆ κ avoiding the measurable cardinals and the forcing to add various long Prikry sequences. The lottery preparation works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, thereby freeing the Laver function concept from the supercompact cardinal context. The Laver preparation [Lav78], which spectacularly makes any supercompact cardinal κ indestructible by <κ-directed closed forcing, has long been an indispensible tool and recognized as an important milestone in large cardinal set theory. Is there such a preparation for the other large cardinals? The Laver preparation does not seem to work with strongly compact and other cardinals. While strong cardinals are successfully treated in [GitShl89], the fundamental lifting tools currently available fail outrightly when applied to strongly compact non-supercompact cardinals. The technology has simply not been available to make strongly compact cardinals even partly indestructible. The extent of our ignorance is made strikingly plain by the fact that the following question has remained open: My research has been supported in part by a grant from the PSC-CUNY Research Foundation, by a CUNY Scholar Incentive Award and by a research fellowship from the Japan Society for the Promotion of Science. I am grateful to my gracious hosts here at Kobe University in Japan, and I would like to thank both Arthur Apter for his insightful suggestions concerning my work with strongly compact cardinals and Tadatoshi Miyamoto for his probing questions on an earlier draft of this paper.