Set Theory: Combinatorics and Large Cardinals

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This project will primarily involve work in two areas. Mitchell hasrecently answered a longstanding question of Shelah concerning theapproachability ideal at the second uncountable cardinal, and thefirst area of research will further exploit these new techniques to betterunderstand this ideal and other questions about closed unboundedsets. A major aim will be to extend these ideas to the successor ofa singular cardinal. The second area of research is in inner modeltheory, where Mitchell will attempt to better understand the situationjust below a Woodin cardinal and at very large cardinals where nocore 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 ofsingular cardinals, such as the first infinite cardinal number whichhas infinitely many smaller infinite cardinals below it. The structure atsuch cardinals is much more difficult than the structure at regular cardinals,and the planned research could contribute significantly to itsunderstanding. Much research in set theory studies different the different possibleforms that the universe of sets might take. It has been shown that, under appropriateassumptions, the "core model" provides an invariant skeleton which shared by all of thesepossible universes, and from which they derive much of theirstructure. The work on inner models will aim at an understanding ofexceptional cases where this skeleton is not yet know to exist.