The Lottery Preparation

Abstract

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 kappa, for example, becomes fully indestructible by kappa-directed closed forcing; a strong cardinal kappa becomes indestructible by less-than-or-equal-kappa-strategically closed forcing; and a strongly compact cardinal kappa becomes indestructible by, among others, the forcing to add a Cohen subset to kappa, the forcing to shoot a club C in kappa which avoids 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.