We show that certain relatively consistent structural properties of the class of supercompact cardinals are also relatively consistent with the Wholeness Axioms.

I analyze the hierarchy of large cardinals between a supercompact cardinal and an almost-huge cardinal. Many of these cardinals are defined by modifying the definition of a high-jump cardinal. A high-jump cardinal is defined as the critical point of an elementary embedding j : V → M such that M is closed under sequences of length sup{ j(f)(κ) | f : κ→ κ }. Some of the other cardinals analyzed i...

Generalizing Woodin’s extender algebra, cf. e.g. [8], we isolate the long extender algebra as a general version of Bukowský’s forcing, cf. [1], in the presence of a supercompact cardinal.

We construct a model containing a proper class of strongly compact cardinals in which no strongly compact cardinal κ is κ supercompact and in which every strongly compact cardinal has its strong compactness resurrectible.

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 κ, for example, becomes fully indestructible b...

Answering a question raised by Luis Pereira, we show that a continuous tree-like scale can exist above a supercompact cardinal. We also show that the existence of a continuous tree-like scale at אω is consistent with Martin’s Maximum.

The paper is concerned with the existence of a universal graph at the successor of a strong limit singular μ of cofinality א0. Starting from the assumption of the existence of a supercompact cardinal, a model

Starting from a supercompact cardinal and a measurable above it, we construct a model of ZFC in which the definable tree property holds at all uncountable regular cardinals. This answers a question from [1]

We prove what the title says. It then follows that zero-dimensional Dugundji space are supercompact. Moreover, their Boolean algebras of clopen subsets turn out to be semigroup algebras.

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 κ, for example, becomes fully indestructible b...