Set Theory: Combinatorics and Large Cardinals

这个项目将主要涉及两个领域的工作。米切尔最近回答了希拉关于第二个不可数基数的可接近性理想的长期问题，第一个研究领域将进一步利用这些新技术来更好地理解这个理想和其他关于封闭无界集的问题。一个主要的目标是将这些想法扩展到一位红衣主教的继任者身上。第二个研究领域是内部模型理论，米切尔将试图更好地理解伍丁基数下的情况，以及目前已知没有核心模型存在的非常大的基数。这些问题涉及集合的基本结构。可接近性理想是由Shelah定义的，用于他对奇异基数的分析，例如第一个无限基数，它下面有无限多个更小的无限基数。这种枢机的结构比普通枢机的结构要困难得多，计划中的研究可以对其理解做出重大贡献。集合论的许多研究都研究了集合集合可能采取的不同可能形式。研究表明，在适当的假设下，“核心模型”提供了一个不变的骨架，所有这些可能的宇宙都共享这个骨架，它们的大部分结构都是从这个骨架中推导出来的。关于内部模型的工作将旨在理解这种骨架尚不存在的特殊情况。

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.