Garti and Shelah [2] state that one can force uκ to be κ+ for supercompact κ with 2κ arbitrarily large, using the technique of Džamonja and Shelah [1]. Here we spell out how this can be done. §

Starting from the existence of many supercompact cardinals, we construct a model of ZFC + GCH in which the tree property holds at a countable segment of successor of singular cardinals.

We force and construct a model in which level by level equivalence between strong compactness and supercompactness holds, along with certain additional “inner model like” properties. In particular, in this model, the class of Mahlo cardinals reflecting stationary sets is the same as the class of weakly compact cardinals, and every regular Jonsson cardinal is weakly compact. On the other hand, w...

Can a supercompact cardinal κ be Laver indestructible when there is a level-by-level agreement between strong compactness and supercompactness? In this article, we show that if there is a sufficiently large cardinal above κ, then no, it cannot. Conversely, if one weakens the requirement either by demanding less indestructibility, such as requiring only indestructibility by stratified posets, or...

The purpose of this communication is to present some recent advances on the consequences that forcing axioms and large cardinals have on the combinatorics of singular cardinals. I will introduce a few examples of problems in singular cardinal combinatorics which can be fruitfully attacked using ideas and techniques coming from the theory of forcing axioms and then translate the results so obtai...

Abstract After reviewing various natural bi-interpretations in urelement set theory, including second-order theories with urelements, we explore the strength of reflection these contexts. Ultimately, prove that abundant atom axiom is bi-interpretable and hence also equiconsistent existence a supercompact cardinal. The proof relies on characterization supercompactness, namely, cardinal $\kappa $...

Given a regular cardinal λ and λ many supercompact cardinals, we describe a type of forcing such that in the generic extension there is a cardinal κ with cofinality λ, the Singular Cardinal Hypothesis at κ fails, and the tree property holds at κ.