CAREER: Forcing and Large Cardinals

集合论的标准公理，即带有选择公理的Zermelo-Fraenkel集合论（ZFC），并不能决定许多自然问题。例如，连续统假设（即不存在基数严格介于整数和实数之间的集合）是独立于标准公理的，1963年Paul Cohen通过强迫的突破方法证明了这一点。从那时起，集合理论中一个长期的项目就是使用强迫来获得相对一致性结果，并研究ZFC公理的强化。这构成了项目的广泛动机：加强ZFC的可能性是什么，相对于ZFC本身施加的约束是什么？该项目的教育部分包括组织两次研讨会，并支持本科生和研究生的研究。ZFC增强的主要候选公理是大基数公理和强强制公理。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，特别是在奇异基数的情况下，用于理解这些扩展的本质，以及我们可以在多大程度上使用强制基数和大基数。Jensen’s square, Shelah’s可接近性属性是“反紧性”类型原则，适用于与l足够相似的模型。相反，树属性是一种类似于大基数属性的反射属性，但可以适用于后继基数。该项目将重点关注这些原则之间的相互作用，以及从大型基数构造的强制扩展。PI还将研究它们如何与单一基数假设和希拉的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”，人们可以根据它来测试新的公理。

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.