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.

这个研究计划涉及组合集合论，这是研究某些无限的数学结构，包括图，树，线性和偏序。这些结构在数学中无处不在，可以从许多角度进行研究。该项目旨在通过扩展对这些结构的基数的理解来帮助回答基本的数学问题。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性、奇异基数组合学和可定义性。 目标包括产生模型，其中许多连续的基数同时具有树属性，通过研究其基数不变量获得关于奇异基数的精确信息，以及构建新的基于扩展的强迫。该奖项反映了NSF的法定使命，并被认为是值得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的支持。

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