Starting from a supercompact cardinal κ, we force and construct a model in which κ is both the least strongly compact and least supercompact cardinal and κ’s strong compactness, but not its supercompactness, is indestructible under arbitrary κ-directed closed forcing.

We construct a model in which there are no @n-Aronszajn trees for any nite n 2, starting from a model with innnitely many supercompact cardinals. We also construct a model in which there is no ++-Aronszajn tree for a strong limit cardinal of coonality !, starting from a model with a supercompact cardinal and a weakly compact cardinal above it.

Let κ > ω be a regular cardinal and λ > κ a cardinal. The following partition property is shown to be consistent relative to a supercompact cardinal: For any f : ⋃ n<ω [X] ⊂ → γ with X ⊂ Pκλ unbounded and 1 < γ < κ there is an unbounded Y ⊂ X with |f“[Y ]⊂| = 1 for any n < ω. Let κ be a regular cardinal > ω, λ a cardinal ≥ κ and F a filter on Pκλ. Partition properties of the form Pκλ→ (F)2 (see...

We prove a revised version of Laver’s indestructibility theorem which slightly improves over the classical result. An application yields the consistency of (κ, κ) ։/ (א1,א0) when κ is supercompact. The actual proofs show that ω1-regressive Kurepatrees are consistent above a supercompact cardinal even though MM destroys them on all regular cardinals. This rather paradoxical fact contradicts the ...

We establish two new Easton theorems for the least supercompact cardinal that are consistent with the level by level equivalence between strong compactness and supercompactness. These theorems generalize [1, Theorem 1]. In both our ground model and the model witnessing the conclusions of our theorem, there are no restrictions on the structure of the class of supercompact cardinals.

It is shown that if there exists a supercompact cardinal then every set of reals, which is an element of L(R), is the projection of a weakly homogeneous tree. As a consequence of this theorem and recent work of Martin and Steel [Martin, D. A. & Steel, J. R. (1988) Proc. Natl. Acad. Sci. USA 85, 6582-6586], it follows that (if there is a supercompact cardinal) every set of reals in L(R) is deter...

An inaccessible cardinal κ is supercompact when (κ, λ)-ITP holds for all λ ≥ κ. We prove that if there is a model of ZFC with two supercompact cardinals, then there is a model of ZFC where simultaneously (א2, μ)-ITP and (א3, μ′)-ITP hold, for all μ ≥ א2 and μ′ ≥ א3.

We present a new forcing notion combining diagonal supercompact Prikry focing with interleaved extender based forcing. We start with a supercompact cardinal κ. In the final model the cofinality of κ is ω, the singular cardinal hypothesis fails at κ and GCH holds below κ. Moreover we define a scale at κ, which has a stationary set of bad points in the ground model.