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

这个项目探讨了逻辑和数学基础三个不同领域的问题。首先，该项目将使用Zermelo-Frenkel集合理论作为基础来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透镜来理解无穷组合学。第二，这个项目将研究集合论和模型论的抽象观点之间的联系。模型论中的许多断言原来都有一定的理论内容，研究者将继续研究什么时候有这种联系，什么时候没有。第三，这个项目将探索描述图组合学的相对较新的领域，以及它与几何和测度论中经典问题的联系。从无穷组合学，该项目将集中在树属性，这是一个紧凑性原则，抽象的科尼格无穷引理更高的基数。该项目的一个具体目标是探索猜想，模的一致性大基数最少不可数基数是唯一的基数，可证明进行反例树的性质。该项目还将探讨其他类似问题。从模型论，调查员将研究模型理论的概念连接到抽象的基本类的分类理论及其集合论的内容。特别感兴趣的属性是驯服性和局部性，分类性和Hanf数。从描述图组合学，研究者将探讨匹配和着色在可定义的图形。这项研究已经被证明有有趣的连接与问题的几何，例如Marczewski的问题是否巴拿赫-塔斯基悖论是可能使用Baire可测量件。

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.