We show relative to strong hypotheses that patterns of compact cardinals in the universe, where a compact cardinal is one which is either strongly compact or supercompact, can be virtually arbitrary. Specifically, we prove if V “ZFC + Ω is the least inaccessible limit of measurable limits of supercompact cardinals + ƒ : Ω → 2 is a function”, then there is a partial ordering P V so that for , There is a proper class of compact cardinals + If (...) ƒ = 0, then the αth compact cardinal is not supercompact + If ƒ = 1, then the αth compact cardinal is supercompact”. We then prove a generalized version of this theorem assuming κ is a supercompact limit of supercompact cardinals and ƒ : κ → 2 is a function, and we derive as corollaries of the generalized version of the theorem the consistency of the least measurable limit of supercompact cardinals being the same as the least measurable limit of nonsupercompact strongly compact cardinals and the consistency of the least supercompact cardinal being a limit of strongly compact cardinals. (shrink)

The Gap Forcing Theorem, a key contribution of this paper, implies essentially that after any reverse Easton iteration of closed forcing, such as the Laver preparation, every supercompactness measure on a supercompact cardinal extends a measure from the ground model. Thus, such forcing can create no new supercompact cardinals, and, if the GCH holds, neither can it increase the degree of supercompactness of any cardinal; in particular, it can create no new measurable cardinals. In a crescendo of what I call (...) exact preservation theorems, I use this new technology to perform a kind of partial Laver preparation, and thereby finely control the class of posets which preserve a supercompact cardinal. Eventually, I prove the ‘As You Like It’ Theorem, which asserts that the class of κ -directed closed posets which preserve a supercompact cardinal κ can be made by forcing to conform with any pre-selected local definition which respects the equivalence of forcing. Along the way I separate completely the levels of the superdestructibility hierarchy, and, in an epilogue, prove that the notions of fragility and superdestructibility are orthogonal — all four combinations are possible. (shrink)

The Lévy-Solovay Theorem [8] limits the kind of large cardinal embeddings that can exist in a small forcing extension. Here I announce a generalization of this theorem to a broad new class of forcing notions. One consequence is that many of the forcing iterations most commonly found in the large cardinal literature create no new weakly compact cardinals, measurable cardinals, strong cardinals, Woodin cardinals, strongly compact cardinals, supercompact cardinals, almost huge cardinals, huge cardinals, and so on.