Singular Combinatorics

奇异组合学

• 批准号: 1362485
• 负责人: Dima Sinapova
• 金额: $ 14.94万
• 依托单位: University of Illinois at Chicago
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-05-01 至 2018-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1362485&HistoricalAwards=false
• 关键词: Singular; Combinatorics

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.

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，哥德尔和科恩表明，连续统假设（即不存在其基数严格介于整数和实数之间的集合）独立于该公理系统。从那时起，集合论中的一个长期项目就是寻找公理的“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。该项目探索组合集合论的各个方面。主要目标是研究大基数、强迫和平方、树性质等原理以及 Shelah 的可能共尾性理论及其在奇异组合数学中的应用之间的相互作用。这项工作是一个项目的一部分，该项目旨在确定奇异基数及其后继者在大基数或强强迫公理的 ZFC 扩展中存在的规范结构。长期目标是了解相对于大基数而言什么是可能的，作为大基数的残余可以获得什么，并发展某些强迫偏序集的理论。强制用于测试 ZFC 的这些增强功能的威力和局限性，而树属性等组合原则提供了关键的测试问题。