  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.
