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，哥德尔和科恩证明了连续统假设（即不存在基数严格介于整数和真实的数之间的集合）与这个公理系统无关。从那时起，集合论中一个长期存在的项目就是找到公理的“正确”加强。有几个候选人，这个项目有助于理解这些扩展的性质。该项目探索组合集合论的各个方面。主要目标是调查大基数之间的相互作用，迫使，和原则，如广场，树的财产，和谢拉的理论可能的共尾性及其应用奇异组合。这项工作是一个项目的一部分，以确定存在于奇异的基数和他们的继任者在ZFC的扩展大基数或强强制公理的规范结构。长期的目标是了解什么是可能的相对于大基数，什么可以得到作为残余的大基数，并发展理论的某些强制偏序集。强迫被用来测试ZFC的这些增强的力量和局限性，而像树属性这样的组合原则提供了关键的测试问题。

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.