Set Theory: Combinatorics and Large Cardinals

该项目将主要涉及两个领域的工作。 Mitchell最近回答了Shelah关于第二不可数基数上的可接近性理想的一个长期存在的问题，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。 一个主要的目标将是扩大这些想法的继任者ofa奇异的红衣主教。 研究的第二个领域是内在模型理论，米切尔将试图更好地理解在Woodin基数之下的情况，以及目前已知不存在核心模型的非常大的基数。这些问题涉及集合宇宙的基本结构。可接近性理想是由Shelah定义的，用于他对奇异基数的分析，比如第一个无穷基数下面有无穷多个更小的无穷基数。 这种枢机的结构比普通枢机的结构要困难得多，计划中的研究可能会大大有助于理解它。集合论中的许多研究都是研究集合论域可能采取的不同形式。已经表明，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。 内部模型的工作将旨在理解这种骨架还不知道存在的例外情况。

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.