Infinitary Combinatorics

集合论是研究数学基础的学科。许多自然的数学问题是独立于通常的数学公理。最著名的例子是连续统假设（CH），它是这样一个陈述：任何真实的数的无限集合要么是可数的，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CH不能被标准的数学公理所反驳，称为ZFC（Zermelo-Fraenkel set theory with the axiom of choice）。然后在1964年，保罗·科恩发明了强迫的方法，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。自科恩的工作以来，现代集合论研究了ZFC约束（即“什么是必要的”）与通过强迫获得的相对一致性结果（即“什么是可能的”）。这两个问题都可以通过无穷组合学来解决，无穷组合学是数学中对无限对象的研究。这项研究为研究生提供了许多项目和研究培训机会。PI将专注于使用强迫和大基数来研究无限对象的属性。主要的动机是分析ZFC约束一致性的结果。该项目将集中分析组合原理，如树的性质，平稳反射和平方原理，以及它们与基数运算，特别是奇异基数运算的关系。树的性质和固定反射是紧型性质，遵循大基数。紧性是这样一种现象，如果某个性质对给定对象的每个较小的子结构都成立，那么它对对象本身也成立。另一方面，正方形性质是在哥德尔的可构造的宇宙L中成立的不确定性的典型实例，并且与大基数不一致。该奖项反映了NSF的法定使命，并被认为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来支持。

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.