Grigor Sargsyan

Descriptive inner model theory

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 tools from both areas. There are several interlaced problems that lie on the border of these two areas of set theory, but one that has been rather central for almost two decades is the conjecture known as the Mouse Set Conjecture (MSC). One particular motivation for resolving MSC is that it provides grounds for solving the inner model problem which dates back to 1960s. There have been some new partial results on MSC and the methods used to prove the new instances suggest a general program for solving the full conjecture. It is then our goal to communicate the ideas of this program to the community at large.

THE MOUSE SET CONJECTURE FOR SETS OF REALS

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 mice over R. Such a representation of models of AD+ is important in core model induction applications. One of the central open problems in descriptive inner model theory is the conjecture known as the Mouse Set Conjecture (MSC). It conjectures that under AD ∗2000 Mathematics Subject Classifications: 03E15, 03E45, 03E60. †

Indestructible strong compactness but not supercompactness

Abstract 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.

On HOD-supercompactness

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

Non-tame mice from tame failures of the unique branch hypothesis y

In this paper, we show that the failure of the unique branch hypothesis (UBH) for tame trees (see \rdef{tame iteration tree}) implies that in some homogenous generic extension of

On the indestructibility aspects of identity crisis

V

Identity crises and strong compactness III: Woodin cardinals

there is a transitive model

VARSOVIAN MODELS I

M

Universal indestructibility for degrees of supercompactness and strongly compact cardinals

containing

Can a large cardinal be forced from a condition implying its negation

Ord \cup \mathbb{R}