Generalizations of the Ultrapower Axiom

在20世纪之交，数学家们发现了一系列逻辑悖论，迫使他们重新评估这门学科的基础。在接下来的几十年里，Zermelo-Frankel集合理论公理（ZFC）作为一个问题的答案出现了：数学的基本假设是什么？从ZFC的九个公设中，可以推导出所有已知的数学定理。尽管逻辑取得了这样的胜利，但一个主要问题仍然存在：有一些数学问题无法单独假设ZFC公理来解决。最著名的是，哥德尔和科恩表明，从这些公理开始，不可能证明或反驳康托尔的连续统假设。现代集合论的主要目标之一是分析和分类超越ZFC的公理系统，这些系统足够强大，可以回答这些不可确定的问题。本课题通过挖掘集合论内部模型中的大基数结构，研究了一个生成集合论公理的框架。该项目涉及学生培训和会议组织，将对数学哲学产生影响。ZFC可以看作是对所有集合的类的结构进行公理化的一种尝试。它是不完整的，因为我们对任意集合的数学直觉不足以决定它们的所有性质。为了解决这个问题，我们可以将注意力限制在集合的较小子类上，这些子集在某种程度上是规范的。Gödel发现有些子类足够丰富，可以满足ZFC公理，但又足够约束，可以确定它们的所有属性。这样的子类称为规范内部模型。例如，在规范的内部模型中，连续统假设是正确的。通过考虑在规范内部模型中持有的语句，可以获得ZFC的扩展。其中一个说法是超功率公理，由PI在他的论文中确定和研究。该公理在无限基数或大基数的层次结构的上游施加了丰富的结构，这是集合论研究的中心对象之一。利用Woodin最近的突破，PI能够制定超功率公理的强推广，该项目提出的研究希望他们将进一步阐明内模型理论和大基数中的主要开放问题。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。

At the turn of the 20th century, mathematicians discovered a series of logical paradoxes that forced them to reevaluate the very foundation of the subject. The Zermelo-Frankel axioms of set theory (ZFC) emerged in the decades that followed as an answer to the question: what are the basic assumptions of mathematics? From the nine postulates of ZFC, one can derive all known theorems of mathematics. Despite this triumph of logic, a major problem remains: there are mathematical problems that cannot be solved assuming the ZFC axioms alone. Most famously, Godel and Cohen showed that starting with these axioms, it is impossible to prove or refute Cantor's Continuum Hypothesis. One of the main goals of modern set theory is to analyze and classify axiomatic systems beyond ZFC that are strong enough to answer these undecidable questions. This project studies a framework for generating set-theoretic axioms by mining the structure of large cardinals in inner models of set theory. This project involves student training and conference organization and will have an impact on the philosophy of mathematics.ZFC can be seen as an attempt to axiomatize the structure of the class of all sets. It is incomplete because our mathematical intuitions about arbitrary sets do not suffice to determine all their properties. To get around this problem, one can restrict attention to smaller subclasses of sets that are somehow canonical. Gödel discovered that there are subclasses that are rich enough to satisfy the ZFC axioms yet constrained enough that all their properties can be determined. Such a subclass is called a canonical inner model. For example, in canonical inner models, the Continuum Hypothesis is true. One obtains extensions of ZFC by considering the statements that hold in canonical inner models. One such statement is the Ultrapower Axiom, identified and studied by the PI in his dissertation. The axiom imposes a rich structure on the upper reaches of the hierarchy of infinite cardinals, or large cardinals, one of the central objects of study in set theory. Taking advantage of a recent breakthrough of Woodin, the PI was able to formulate strong generalizations of the Ultrapower Axiom, which this project proposes to study in hopes that they will shed further light on major open problems in inner model theory and large cardinals.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.