Infinitary Combinatorics

集合论是对数学基础的研究。许多自然数学问题不依赖于通常的数学公理。最著名的例子是连续统假设（CH），它是一种陈述，即任何无限实数集合要么是可数的，要么与所有实数具有相同的大小。这就是希尔伯特的第一个问题。第一个突破是在1940年由库尔特·哥德尔（Kurt Godel）提出的，他证明了CH不能被称为ZFC （Zermelo-Fraenkel集合理论与选择公理）的标准数学公理所反驳。然后在1964年Paul Cohen发明了强迫的方法并用它来证明连续统假设实际上是独立于ZFC的。换句话说，CH和它的否定都不是ZFC公理的逻辑推论。自Cohen的工作以来，现代集合理论研究了ZFC约束(即。“什么是必要的”)与通过强制获得的相对一致性结果(即：“什么是可能的”)。这两个问题都是由无穷组合学来解决的，无穷组合学是对数学中无限对象的研究。这项研究为研究生提供了许多项目和研究训练机会。PI将专注于使用强制和大基数来研究无限对象的属性。主要动机是根据一致性结果分析zfc约束。该项目将集中分析组合原理，如树性质，静止反射和平方原理，以及它们与基数算术，特别是奇异基数算术的关系。树属性和静止反射是紧致性类型的属性，它们来自于大基数。紧致性是一种现象，如果某一特性适用于给定物体的每一个较小的子结构，那么它也适用于该物体本身。另一方面，平方性质是不紧性的典型实例，它在哥德尔的可构造宇宙L中成立，与大基数不一致。本项目将分析它们的相互作用以及与基数算术的相互作用。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Set theory is the study of the foundations of mathematics. Many natural mathematical questions are independent of the usual mathematical axioms. The most famous example is the continuum hypothesis (CH), which is the statement that any infinite set of real numbers is either countable or has the same size as all the reals. This became Hilbert's First Problem. The first breakthrough was in 1940 by Kurt Godel, who showed that CH cannot be refuted by the standard mathematical axioms, known as ZFC (Zermelo-Fraenkel set theory with the axiom of choice). Then in 1964 Paul Cohen invented the method of forcing and used it to show that the continuum hypothesis is actually independent of ZFC. In other words, neither CH, not its negation is a logical consequence of the ZFC axioms. Since Cohen's work, modern set theory investigates ZFC constraints (i.e. "what is necessary") versus relative consistency results obtained by forcing (i.e. "what is possible"). Both questions are addressed by infinitary combinatorics, the study of infinite objects in mathematics. This study generates many projects and research training opportunities for graduate students.The PI will focus on using forcing and large cardinals to investigate properties of infinite objects. The main motivation is analyzing ZFC-constraints against consistency results. The project will center on analyzing combinatorial principles, such as the tree property, stationary reflection and square principles, and their relation to cardinal arithmetic, especially singular cardinal arithmetic. The tree property and stationary reflection are compactness type properties that follow from large cardinals. Compactness is the phenomenon where if a certain property holds for every smaller substructure of a given object, then it holds for the object itself. On the other hand, square properties are canonical instances of incompacntess that hold in Godel's constructible universe L and are at odds with large cardinals. This project will analyze their interplay and interaction with cardinal arithmetic.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.