CAREER: Forcing and Large Cardinals

集合论的标准公理，Zermelo-Fraenkel集合论与选择公理（ZFC），并没有决定许多自然问题。例如，连续统假设（即不存在基数严格介于整数基数和真实的数基数之间的集合）是独立于标准公理的，正如保罗·科恩在1963年通过强迫突破方法所证明的那样。 从那时起，集合论中的一个长期项目一直是使用相对一致性结果的强制，并研究ZFC公理的加强。这构成了该项目的广泛动机：什么是可能的，在加强ZFC，与什么限制是由ZFC本身施加？该项目的教育部分的特点是组织两个研讨会和支持本科生和研究生的研究。ZFC加强的主要候选人是大基数公理和强强迫公理。 强制使用大基数的模型也是显示一致性结果的最强大工具。组合原理，特别是在奇异基数上，被用来理解这些扩展的性质，以及我们可以对强迫和大基数做多少事情。詹森的平方，谢拉的可接近性性质是“反紧”型原则，在充分类似L的模型中成立。相反，树属性是一个反射属性，类似于大型基数属性，但可以在后继基数上保持。该项目将集中在这些原则之间的相互作用，并迫使从大基数构造的扩展。PI还将研究它们如何与奇异基数假设和Shelah的PCF理论相互作用。后者主要是在ZFC中决定的，并提供了某些“规范不变量”，人们可以测试新的公理。

The standard axioms of set theory, Zermelo-Fraenkel set theory with the axiom of choice (ZFC), do not decide many natural questions. For example, the Continuum Hypothesis (that there is no set whose cardinality is strictly between that of the integers and that of the real numbers) is independent of the standard axioms, as shown in 1963 by Paul Cohen through the breakthrough method of forcing. Since then, a longstanding project in set theory has been to use forcing for relative consistency results and to study strengthening of the ZFC axioms. This constitutes the broad motivation of the project: What is possible in strengthening of ZFC, versus what constraints are imposed by ZFC itself? The educational component of the project features organizing two workshops and support for undergraduate and graduate student research.The main candidates for ZFC strengthenings are large cardinal axioms and strong forcing axioms. Forcing over a model with large cardinals is also the most powerful tool for showing consistency results. Combinatorial principles, especially at singular cardinals are used to understand both the nature of these extensions, and how much we can do with forcing and large cardinals. Jensen's square, Shelah's approachability property are "anti-compactness" type principles that hold in models that sufficiently resemble L. In contrast, the tree property is a reflection property that resembles large cardinal properties, but can hold at successor cardinals. The project will focus on the interplay between these principles and forcing extensions constructed from large cardinals. The PI will also investigate how they interact with the singular cardinal hypothesis and Shelah's PCF theory. The latter is mostly decided in ZFC, and provides certain "canonical invariants," against which one can test new axioms.