The lottery preparation

The lottery preparation, a new general kind of Laver preparation, works uni formly with supercompact cardinals, strongly compact cardinals, strong card inals, measurable cardinals, or what have you. And like the Laver preparati on, the lottery preparation makes these cardinals indestructible by various kinds of further forcing. A supercompact cardinal kappa, for example, beco mes fully indestructible by < kappa-directed closed forcing; a strong cardi nal kappa becomes indestructible by less than or equal to kappa-strategical ly closed forcing; and a strongly compact cardinal k becomes indestructible by, among others, the forcing to add a Cohen subset to kappa, the forcing to shoot a club C subset of or equal to kappa avoiding the measurable cardi nals and the forcing to add various long Prikry sequences. The lottery prep aration works best when performed after fast function forcing, which adds a new completely general kind of Laver function for any large cardinal, ther eby freeing the Laver function concept from the supercompact cardinal conte xt. (C) 2000 Elsevier Science B.V. All rights reserved.