Combinatorics of Filters, Large Cardinal and Prikry-Type Forcing

滤波器、大基数和 Prikry 型强迫的组合

• 批准号: 2346680
• 负责人: Tom Benhamou
• 金额: $ 15万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2023
• 资助国家: 美国
• 起止时间: 2023-08-15 至 2026-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2346680&HistoricalAwards=false
• 关键词: Combinatorics; Filters; Large; Cardinal; Prikry

Our understanding of the mathematical universe, or more precisely, the behavior of the infinite, is quite limited. The main reason for this limitation is the incapability of the Zermelo-Fraenkel axiomatic system of set theory (ZFC), which is the global standard for the formal foundations of mathematics, to determine basic questions about the infinite. This channeled the research in modern set theory to mainly two directions. The first direction is the search for new restraints imposed by ZFC on the behavior of the infinite. This direction had great success in the realm of singular cardinal arithmetic where new surprising principles were discovered by Silver and later by Shelah. This project aims to study some of these principles. The second direction is the subtle interaction between extensions of ZFC to stronger axiomatic systems and statements which are unsettled by ZFC. Perhaps the most prominent axiomatic systems are extensions of ZFC by the so-called large cardinal axioms. This project contributes to the development of new constructions with large cardinals, implementing them to analyze the interaction between those large cardinal axioms and unsettled statements. The PI will lean of his extensive experiences of mentoring underprivileged students in his teaching activities.This project deals with several central areas in set theory: (i) forcing theory and more particularly Prikry-type forcing; (ii) cardinal arithmetic; and (iii) infinitary combinatorics. Forcing with a Prikry-type forcing notion is perhaps the most important technique to generate models with non-trivial patterns of singular cardinal arithmetic. This project contributes to the investigation and sophistication of these techniques through several aspects: the development of combinatorics of ultrafilters, the discovery of new connections of this theory with other areas of set theory such as inner model theory and infinite combinatorics, the characterization of intermediate models of several Prikry-type models such as the Magidor-Radin model and the tree Prikry forcing, and obtaining stationary reflection at the successor of the first singulars of uncountable cofinalities. One particularly interesting combinatorial property of ultrafilters which is investigated in this project is the Galvin property which recently gained renewed interest due to the surprising connections of this property to the structure of ultrafilters in canonical inner models, the Tukey order, Prikry forcing, partition relations and more.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

我们对数学宇宙的理解，或者更准确地说，对无限行为的理解是相当有限的。这种限制的主要原因是集合论（ZFC）的策梅洛-弗兰克尔公理系统（ZFC）无法确定有关无限的基本问题，而ZFC是数学形式基础的全球标准。这将现代集合论的研究主要引导到两个方向。第一个方向是寻找 ZFC 对无限行为施加的新限制。这个方向在奇异基数算术领域取得了巨大成功，西尔弗和后来的谢拉发现了新的令人惊讶的原理。该项目旨在研究其中一些原则。第二个方向是 ZFC 向更强的公理系统的扩展和 ZFC 未解决的陈述之间的微妙相互作用。也许最突出的公理系统是所谓的大基数公理对 ZFC 的扩展。该项目有助于开发具有大基数的新结构，实施它们来分析这些大基数公理和未解决的陈述之间的相互作用。 PI 将在教学活动中借鉴他指导贫困学生的丰富经验。该项目涉及集合论的几个核心领域：(i) 强迫理论，更具体地说是 Prikry 型强迫； (ii) 基数算术； (iii) 无限组合学。使用 Prikry 型强迫概念进行强迫可能是生成具有奇异基数算术的非平凡模式的模型的最重要技术。该项目通过几个方面为这些技术的研究和完善做出了贡献：超滤器组合学的发展、该理论与集合论其他领域（如内模型理论和无限组合学）的新联系的发现、中间模型的表征多个 Prikry 型模型，例如 Magidor-Radin 模型和树 Prikry 强迫，并在不可数共尾性的第一个奇点的后继处获得平稳反射。本项目研究的超滤器的一个特别有趣的组合特性是加尔文特性，由于该特性与规范内模型中的超滤器结构、Tukey 阶、Prikry 强迫、分区关系和更多。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。