Failures of GCH and the level by level equivalence between strong compactness and supercompactness

Abstract

Key words Supercompact cardinal, strongly compact cardinal, strong cardinal, measurable cardinal, nonreflecting stationary set of ordinals, level by level equivalence between strong compactness and supercompactness. MSC (2000) 03E35, 03E55We force and obtain three models in which level by level equivalence between strong compactness and supercompactness holds and in which, below the least supercompact cardinal, GCH fails unboundedly often. In two of these models, GCH fails on a set having measure 1 with respect to certain canonical measures. There are no restrictions in all of our models on the structure of the class of supercompact cardinals.

Supercompactness and measurable limits of strong cardinals II: Applications to level by level equivalence

“…Finally, and perhaps most importantly, in Theorem 1.1, it can be the case that 2 κ > κ ++ . This contrasts sharply with the results found in [2] and [1], where the structure of the class of supercompact cardinals can be arbitrary, level by level equivalence between strong compactness and supercompactness holds, yet if GCH fails at the least supercompact cardinal κ, 2 κ = κ ++ .…”

“…Nonetheless, it has been shown (see [1,2,4,5,6]) that this property is consistent with exotic and unusual occurrences that one would not expect in an inner model, such as significant failures of GCH. It is thus reasonable to ask how far these kinds of situations can be extended.…”

An Easton theorem for level by level equivalence

“…In [3,5,1], results concerning the possible failures of GCH that are consistent with the level by level equivalence between strong compactness and supercompactness were established. We refer readers to these papers for further details on the kinds of theorems proven.…”

“…Before presenting the proofs of our theorems, we briefly state some preliminary information. Our notation and terminology will follow that given in [1]. We do wish to mention a few things explicitly, however.…”

Indestructibility under adding Cohen subsets and level by level equivalence

“…1 Note that earlier indestructibility theorems for the least supercompact cardinal in conjunction with level by level equivalence between strong compactness and supercompactness may be found in the papers [11], [1], [7], and [5].…”