We show that the Mouse Set Conjecture for sets of reals is true in the minimal model of ADR + “Θ is regular”. As a consequence, we get that below ADR + “Θ is regular”, models of AD + +¬ADR are hybrid… (More)

Building on the work of Schimmerling ([8]) and Steel ([14]), we show that the failure of square principle at a singular strong limit cardinal implies that there is a non-tame mouse. The proof… (More)

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees (see Definition 5.1) implies that in some homogenous generic extension of V there is a transitive model M… (More)

We introduce a covering conjecture and show that it holds below ADR + “Θ is regular”. We then use it to show that in the presence of mild large cardinal axioms, PFA implies that there is a transitive… (More)

During his Fall 2005 set theory seminar, Woodin asked whether Vsupercompactness implies HOD-supercompactness. We show, as he predicted, that that the answer is no.

The purpose of this paper is to outline some recent progress in descriptive inner model theory, a branch of set theory which studies descriptive set theoretic and inner model theoretic objects using… (More)

We show that it is consistent, relative to n ∈ ω supercompact cardinals, for the strongly compact and measurable Woodin cardinals to coincide precisely. In particular, it is consistent for the first… (More)

The main contribution of this paper is a the analysis of HOD below the theory AD+ + “The largest Suslin cardinal is a member of the Solovay sequence”. It is shown that below the aforementioned… (More)

Working under AD, we investigate the length of prewellorderings given by the iterates of M2k+1, which is the minimal proper class mouse with 2k + 1 many Woodin cardinals. In particular, we answer… (More)