Combinatorial Set Theory, Forcing, and Large Cardinals

数学中的许多自然问题不能仅用数学的标准公理来回答。最著名的例子是连续统假设，它认为实数的任意无限子集要么是可数的，要么与整个实数集一一对应。这种被称为强迫的技术可以用来证明连续统假设在逻辑上独立于数学公理的标准集合，即Zermelo-Fraenkel集合理论与选择公理（ZFC）。现代集合论的一个主要主题是相对一致性结果和zfc增强的发展。zfc增强是由大基数假设给出的，它断言某些高度紧致的数学对象，称为大基数，具有强反射性质，也就是说，如果一个性质在大基数上成立，它必须在它下面的许多基数上成立。假设存在较大的基数，可以使用强迫方法创建各种数学模型。这个项目分析了这些结构，动机是两个互补的概念：什么是数学上必要的？什么在数学上是充分的？本项目为研究生提供研究训练机会。该项目的主要目标是分析大型基数和强制的可能性，以及ZFC施加的约束。在存在大基数的情况下强制执行是创建ZFC模型并证明一致性结果的主要工具。与具有大基数相比，Gödel的可构造宇宙L是集合论的最小类模型。两个核心问题是：宇宙在多大程度上与L相似，以及通过强制可以达到多大程度的紧致度。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。紧凑性是由一些关键的组合原则（如树属性及其增强）捕获的。另一方面，正方形属性是不紧凑性的典型实例，并作为模型与l的接近程度的标尺。该项目将重点关注这些原则之间的相互作用，以及从大基数构建的强制扩展。PI还将研究它们如何与基数算术，特别是与奇异组合相互作用。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。

Many natural questions in mathematics cannot be answered in terms of the standard axioms of mathematics alone. The most famous example is the Continuum Hypothesis, which states that any infinite subset of the real numbers is either countable or in a one-to-one correspondence with the whole set of real numbers. The technique known as forcing can be used to show that the Continuum Hypothesis is logically independent of the standard set of axioms for mathematics, known as Zermelo-Fraenkel set theory with the axiom of choice (ZFC). A major theme in modern set theory is development of relative consistency results and ZFC-strengthenings. ZFC-strengthenings are given by large cardinal hypotheses, which assert the existence of certain highly compact mathematical objects, called large cardinals, with strong reflection properties, that is, if a property holds at the large cardinal, it must hold at many cardinals below it. Assuming large cardinals exist, the method of forcing can be used to create various mathematical models. This project analyzes these constructions, motivated by two complementary notions: what is mathematically necessary? what is mathematically sufficient? The project provides research training opportunities for graduate students.The main objectives of the project are analyzing what is possible from large cardinals and forcing, versus what constraints are imposed by ZFC. Forcing in the presence of large cardinals is the main tool to create models of ZFC and prove consistency results. In contrast to having large cardinals, Gödel's constructible universe L is the minimal class model of set theory. Two central questions are how much the universe resembles L, and how much compactness can be achieved by forcing. 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. It follows from large cardinals, and often fails in L. Compactness is captured by some key combinatorial principles such as the tree property and its strengthenings. On the other hand, square properties are the canonical instance of incompactness and serve as a yardstick how close a model is to L. The project will focus on the interplay between these principles and forcing extensions constructed from large cardinals. The PI will also investigate how they interact with cardinal arithmetic and especially with singular combinatorics.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.