Singular Combinatorics

事实证明，集合论的标准公理并不能解决许多经典问题。例如，Gödel和Cohen证明了连续统假设(没有基数严格介于整数和实数之间的集合)独立于这个公理系统。从那时起，集合论的一个长期项目就是找到公理的“正确”强化。有几个候选者，这个项目有助于理解这些扩展的性质。这个项目探索了组合集合论的各个方面。主要目的是研究大基数、强迫和诸如平方、树性质和谢拉的可能余定性理论以及它们在奇异组合学中的应用等原理之间的相互作用。这项工作是一个项目的一部分，该项目旨在通过大基数或强强迫公理来确定存在于奇异基数及其后继者的ZFC扩展的正则结构。长期目标是了解相对于大基数有什么可能，作为大基数的残余物可以得到什么，并发展某些强迫偏序集理论。强迫被用来测试ZFC的这些强化的威力和局限性，而像树性质这样的组合原理提供了关键的测试题。

It turns out that standard axioms of set theory do not settle many classical questions. For example, Gödel and Cohen showed that 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 this axiom system. Since then, a long standing project in set theory has been to find the "right" strengthening of the axioms. There are several candidates, and this project contributes to understanding of the nature of these extensions. This project explores various aspects of combinatorial set theory. The main goal is to investigate the interplay between large cardinals, forcing, and principles such as square, the tree property, and Shelah's theory of possible cofinalities and their applications to singular combinatorics. The work is part of a project to determine the canonical structures that exist at singular cardinals and their successors in extensions of ZFC by large cardinals or strong forcing axioms. The long term goal is understanding what is possible relative to large cardinals, what can be obtained as remnants of large cardinals, and developing the theory of certain forcing posets. Forcing is used to test both the power and limitations of these strengthenings of ZFC, and combinatorial principles like the tree property provide the key test questions.