Set Theory: Combinatorics and Large Cardinals

University Of Florida, Gainesville FL

Investigators

Abstract

This project will primarily involve work in two areas. Mitchell has recently answered a longstanding question of Shelah concerning the approachability ideal at the second uncountable cardinal, and the first area of research will further exploit these new techniques to better understand this ideal and other questions about closed unbounded sets. A major aim will be to extend these ideas to the successor of a singular cardinal. The second area of research is in inner model theory, where Mitchell will attempt to better understand the situation just below a Woodin cardinal and at very large cardinals where no core model is presently known to exist. These questions concern the basic structure of the universe of sets. The approachability ideal was defined by Shelah for use in his analysis of singular cardinals, such as the first infinite cardinal number which has infinitely many smaller infinite cardinals below it. The structure at such cardinals is much more difficult than the structure at regular cardinals, and the planned research could contribute significantly to its understanding. Much research in set theory studies different the different possible forms that the universe of sets might take. It has been shown that, under appropriate assumptions, the "core model" provides an invariant skeleton which shared by all of these possible universes, and from which they derive much of their structure. The work on inner models will aim at an understanding of exceptional cases where this skeleton is not yet know to exist.