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。Zermelo-Frankel集合论公理(Zfc)在随后的几十年里出现，作为对这个问题的答案：数学的基本假设是什么？从ZFC的九个公设中，我们可以推导出所有已知的数学定理。尽管逻辑取得了这一胜利，但一个主要问题仍然存在：仅凭ZFC公理无法解决一些数学问题。最著名的是，戈德尔和科恩证明，从这些公理出发，不可能证明或驳斥康托的连续统假设。现代集合论的主要目标之一是分析和分类ZFC以外的公理系统，这些公理系统足够强大，足以回答这些无法决定的问题。这个项目研究了一个通过挖掘集合论内部模型中大基数的结构来生成集合论公理的框架。这个项目涉及学生培养和会议组织，并将对数学哲学产生影响。ZFC可以被视为对所有集合的班级结构公理化的尝试。它是不完整的，因为我们关于任意集合的数学直觉不足以确定它们的所有性质。为了绕过这个问题，人们可以将注意力限制在某种程度上规范的集合的较小子类上。Gödel发现，有一些子类丰富到足以满足ZFC公理，但又受到足够的约束，以至于它们的所有性质都可以确定。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。人们通过考虑在规范内模型中成立的语句来获得ZFC的扩展。一个这样的说法是超功率公理，由PI在他的论文中识别和研究。公理将丰富的结构强加于无限基数或大基数的层次结构的上游，这是集合论研究的中心对象之一。利用Woodin最近的突破，PI能够制定出对超功率公理的强有力的概括，本项目建议对其进行研究，希望它们将进一步阐明内部模型理论和大型基数中的重大公开问题。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.