Let κ < λ be regular cardinals. We say that an embedding j : V → M with critical point κ is λ-tall if λ < j(κ) and M is closed under κ-sequences in V . Silver showed that GCH can fail at a measurable cardinal κ, starting with κ being κ++-supercompact. Later, Woodin improved this result, starting from the optimal hypothesis of a κ++-tall measurable cardinal κ. Now more generally, suppose that κ ...

Building on the work of Schimmerling ([11]) and Steel ([17]), we show that the failure of square principle at a singular strong limit cardinal implies that there is a non-tame mouse. This is the first step towards getting a model of ADR + “Θ is regular” from PFA via the core model induction. One of the wholly grails of inner model program has been determining the exact consistency strength of f...

We give an alternate characterization of a combinatorial property of measures on pKX introduced by Menas. We use this characterization to prove that if k is supercompact, then all measures on pKX in a certain class have the partition property. This result is applied to obtain a self-contained proof that if k is supercompact and X is the least measurable cardinal greater than k, then Solovay's "...

The extender based Prikry forcing notion is being generalized to supercompact extenders.

We construct models for the level by level equivalence between strong compactness and supercompactness containing failures of GCH at inaccessible cardinals. In one of these models, no cardinal is supercompact up to an inaccessible cardinal, and for every inaccessible cardinal δ, 2δ > δ++. In another of these models, no cardinal is supercompact up to an inaccessible cardinal, and the only inacce...

Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal κ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2 supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpubl...

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X → LX is an epimorphism for all X and the class of L-local objects is defined by an absolu...

Generalizing some earlier techniques due to the second author, we show that Menas’ theorem which states that the least cardinal κ which is a measurable limit of supercompact or strongly compact cardinals is strongly compact but not 2 supercompact is best possible. Using these same techniques, we also extend and give a new proof of a theorem of Woodin and extend and give a new proof of an unpubl...