Combinatorial Set Theory, Model Theory of Abstract Elementary Classes, and Borel Combinatorics

组合集合论、抽象初等类模型论和 Borel 组合学

• 批准号: 1700425
• 负责人: Itay Neeman
• 金额: $ 11.7万
• 依托单位: University of California-Los Angeles
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2017
• 资助国家: 美国
• 起止时间: 2017-07-01 至 2021-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1700425&HistoricalAwards=false
• 关键词: Combinatorial; Set; Theory; Model; Abstract

This project explores questions in three separate areas of logic and foundations of mathematics. First, the project will study possible behaviors of the mathematical universe using Zermelo-Frenkel set theory as a foundation. The main goal is to understand infinitary combinatorics through the lens of forcing and large cardinals. Second, the project will investigate the connections between set theory and an abstract view of model theory. Many assertions in model theory turn out to have set theoretic content, and the investigator will pursue the question of when there is such a connection and when there is not. Third, the project will explore the relatively recent field of descriptive graph combinatorics and its connections with classical questions in geometry and measure theory. From infinitary combinatorics, the project will focus on the tree property, which is a compactness principle that abstracts the Konig infinity lemma to higher cardinals. A specific goal of the project is to explore the conjecture that modulo the consistency of large cardinals the least uncountable cardinal is the only cardinal which provably carries a counterexample to the tree property. The project will also pursue other similar questions. From model theory, the investigator will study model theoretic notions connected to the classification theory of abstract elementary classes and their set theoretic content. Particular properties of interest are tameness and locality, categoricity, and Hanf numbers. From descriptive graph combinatorics, the investigator will explore matchings and colorings in definable graphs. This study has already been shown to have interesting connections with questions in geometry, for example Marczewski's question about whether the Banach-Tarski paradox is possible using Baire measurable pieces.

该项目探讨逻辑和数学基础三个不同领域的问题。首先，该项目将使用策梅洛-弗兰克尔集合论作为基础，研究数学宇宙的可能行为。主要目标是通过强迫和大基数的视角来理解无限组合学。其次，该项目将研究集合论和模型论抽象观点之间的联系。模型理论中的许多断言最终都具有一定的理论内容，研究者将追究何时存在这种联系、何时不存在这种联系的问题。第三，该项目将探索描述性图组合学的相对较新的领域及其与几何和测度论中经典问题的联系。从无穷组合学来看，该项目将重点关注树属性，这是一种将 Konig 无穷引理抽象为更高基数的紧致性原理。该项目的一个具体目标是探索这样的猜想：以大基数的一致性为模，最小不可数基数是唯一可证明带有树属性反例的基数。该项目还将探讨其他类似问题。从模型理论出发，研究者将研究与抽象基本类的分类理论及其集合理论内容相关的模型理论概念。特别感兴趣的属性是驯服性和局部性、分类性和汉夫数。从描述性图形组合学中，研究者将探索可定义图形中的匹配和着色。这项研究已经被证明与几何问题有有趣的联系，例如 Marczewski 的问题，即巴纳赫-塔斯基悖论是否可以使用贝尔可测量件来实现。

项目成果

On the powersets of singular cardinals in HOD

关于 HOD 中奇异基数的幂集

• DOI: 10.1090/proc/14913
• 发表时间: 2020
• 期刊: Proceedings of the American Mathematical Society
• 影响因子: 1
• 作者: Ben-Neria, Omer;Gitik, Moti;Neeman, Itay;Unger, Spencer
• 通讯作者: Unger, Spencer

STATIONARY REFLECTION

静止反射

• DOI: 10.1017/jsl.2020.64
• 发表时间: 2020
• 期刊: The Journal of Symbolic Logic
• 作者: HAYUT, YAIR;UNGER, SPENCER
• 通讯作者: UNGER, SPENCER

The ineffable tree property and failure of the singular cardinals hypothesis

不可言喻的树性质和奇异基数假说的失败

• DOI: 10.1090/tran/8110
• 发表时间: 2020
• 期刊: Transactions of the American Mathematical Society
• 影响因子: 1.3
• 作者: Cummings, James;Hayut, Yair;Magidor, Menachem;Neeman, Itay;Sinapova, Dima;Unger, Spencer
• 通讯作者: Unger, Spencer

MEASURABLE REALIZATIONS OF ABSTRACT SYSTEMS OF CONGRUENCES

抽象同余系统的可测量实现

• DOI: 10.1017/fms.2020.4
• 发表时间: 2020
• 期刊: Sigma
• 作者: CONLEY, CLINTON T.;MARKS, ANDREW S.;UNGER, SPENCER T.
• 通讯作者: UNGER, SPENCER T.