We establish a new Easton theorem for the least supercompact cardinal κ that is consistent with the level by level equivalence between strong compactness and supercompactness. This theorem is true in any model of ZFC containing at least one supercompact cardinal, regardless if level by level equivalence holds. Unlike previous Easton theorems for supercompactness, there are no limits on the East...

We show that the theories “ZFC + There is a supercompact cardinal” and “ZFC + There is a supercompact cardinal + Level by level inequivalence between strong compactness and supercompactness holds” are equiconsistent.

Let φ1 stand for the statement V = HOD and φ2 stand for the Ground Axiom. Suppose Ti for i = 1, . . . , 4 are the theories “ZFC + φ1 + φ2”, “ZFC + ¬φ1 + φ2”, “ZFC + φ1 + ¬φ2”, and “ZFC + ¬φ1 + ¬φ2” respectively. We show that if κ is indestructibly supercompact and λ > κ is inaccessible, then for i = 1, . . . , 4, Ai =df {δ < κ | δ is an inaccessible cardinal which is not a limit of inaccessible...

We construct a model for the level by level equivalence between strong compactness and supercompactness with an arbitrary large cardinal structure in which the least supercompact cardinal κ has its strong compactness indestructible under κ-directed closed forcing. This is in analogy to and generalizes [3, Theorem 1], but without the restriction that no cardinal is supercompact up to an inaccess...

In this paper we consider two methods for producing models with some global behavior of the continuum function on singular cardinals and the failure of weak square. The first method is as an extension of Sinapova’s work [21]. We define a diagonal supercompact Radin forcing which adds a club subset to a cardinal κ while forcing the failure of SCH everywhere on the club. The intuition from Sinapo...

We assume the existence of a supercompact cardinal and produce a model with weak square but no very good scale at a particular cardinal. This follows work of Cummings, Foreman, and Magidor, but uses a different approach. We produce another model, starting from countably many supercompact cardinals, where κ,<κ holds but κ,λ fails for λ < κ.

We construct a model for the level by level equivalence between strong compactness and supercompactness in which below the least supercompact cardinal κ, there is an unbounded set of singular cardinals which witness the only failures of GCH in the universe. In this model, the structure of the class of supercompact cardinals can be arbitrary.

We show that given ω many supercompact cardinals, there is a generic extension in which there are no Aronszajn trees at אω+1. This is an improvement of the large cardinal assumptions. The previous hypothesis was a huge cardinal and ω many supercompact cardinals above it, in Magidor-Shelah [7].

We show that, in terms of both implication and consistency strength, an extendible with a larger strong cardinal is stronger than an enhanced supercompact, which is itself stronger than a hypercompact, which is itself weaker than an extendible. All of these are easily seen to be stronger than a supercompact. We also study C-supercompactness.