Forcing, Large Cardinals, and Infinitary Combinatorics

强迫、大基数和无限组合

• 批准号: 2054532
• 负责人: James Cummings
• 金额: $ 18万
• 依托单位: Carnegie-Mellon University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2024-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054532&HistoricalAwards=false
• 关键词: Forcing; Large; Cardinals; Infinitary; Combinatorics

This program of research concerns combinatorial set theory, which is the study of certain infinite mathematical structures, including graphs, trees, and linear and partial orders. These structures are ubiquitous in mathematics and can be studied from many points of view. This project aims to help answer fundamental mathematical questions by extending understanding of the cardinality of these structures. The research involves ideas from areas including logic, algebra, and the theory of infinite games.The main topics in the project are compactness properties, singular cardinal combinatorics, and definability. The goals include producing models where many consecutive cardinals simultaneously have the tree property, getting refined information about singular cardinals through the study of their cardinal invariants, and constructing novel extender-based forcings. The tools will include inner model theory, forcing, and possible cofinalities (PCF) theory.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.

该研究项目涉及组合集合论，它是对某些无限数学结构的研究，包括图、树、线性和偏序。这些结构在数学中无处不在，可以从多种角度进行研究。该项目旨在通过扩展对这些结构基数的理解来帮助回答基本数学问题。该研究涉及逻辑、代数和无限博弈理论等领域的思想。该项目的主要主题是紧致性、奇异基数组合和可定义性。 目标包括生成许多连续基数同时具有树属性的模型，通过研究其基数不变量来获取有关奇异基数的精确信息，以及构建新颖的基于扩展器的强制。这些工具将包括内部模型理论、强迫和可能共尾性 (PCF) 理论。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

THE TREE PROPERTY AT THE TWO IMMEDIATE SUCCESSORS OF A SINGULAR CARDINAL

单一红衣主教的两个直接继承人的树属性

• DOI: 10.1017/jsl.2020.11
• 发表时间: 2021
• 期刊: The Journal of Symbolic Logic
• 作者: CUMMINGS, JAMES;HAYUT, YAIR;MAGIDOR, MENACHEM;NEEMAN, ITAY;SINAPOVA, DIMA;UNGER, SPENCER
• 通讯作者: UNGER, SPENCER

Normal measures on large cardinals

大红衣主教的正常措施

• DOI: 10.1090/btran/132
• 发表时间: 2023
• 期刊: Series B
• 作者: Apter, Arthur;Cummings, James
• 通讯作者: Cummings, James