Combinatorial Set Theory, Forcing, and Large Cardinals

数学中的许多自然问题不能仅仅用标准的数学公理来回答。最著名的例子是连续统假设，它指出任何真实的数的无限子集要么是可数的，要么与整个真实的数集一一对应。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的标准公理集，称为Zermelo-Fraenkel集理论与选择公理（ZFC）。 现代集合论的一个主要主题是相对一致性结果和ZFC-强化的发展。ZFC-强化是由大基数假设给出的，它断言存在某些高度紧凑的数学对象，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。假设大基数存在，强迫方法可以用来创建各种数学模型。 这个项目分析了这些结构，由两个互补的概念：什么是数学上必要的？什么是数学上的充分性？该项目为研究生提供研究培训机会。该项目的主要目标是分析大型红雀和强迫可能发生的情况，以及ZFC施加的限制。在大基数存在下的强迫是创建ZFC模型和证明一致性结果的主要工具。与拥有大基数相反，哥德尔的可构造论域L是集合论的最小类模型。两个核心问题是宇宙有多像L，以及通过强迫可以达到多大的紧凑性。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。它遵循大枢机主教，并经常失败的L。紧性是由一些关键的组合原则，如树的性质和它的加强。另一方面，平方性质是非紧性的典型实例，并作为衡量模型与L有多接近的标准。该项目将集中在这些原则之间的相互作用，并迫使从大基数构造的扩展。PI还将研究它们如何与基数算术，特别是奇异组合学相互作用。该奖项反映了NSF的法定使命，并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，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.