Generalizations of the Ultrapower Axiom

超能力公理的概括

• 批准号: 2401789
• 负责人: Gabriel Goldberg
• 金额: $ 40万
• 依托单位: University of California-Berkeley
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2024
• 资助国家: 美国
• 起止时间: 2024-07-01 至 2029-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2401789&HistoricalAwards=false
• 关键词: Generalizations; Ultrapower; Axiom

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.

在世纪之交，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。集合论的泽梅洛-弗兰克尔公理（ZFC）在随后的几十年中出现，作为对以下问题的回答：数学的基本假设是什么？从ZFC的九个公设，人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：有些数学问题不能仅仅假设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，这些公理系统足够强大，可以回答这些不可判定的问题。本项目研究一个通过挖掘集合论内部模型中的大基数结构来生成集合论公理的框架。该项目涉及学生培训和会议组织，并将对数学哲学产生影响。ZFC可以被视为试图公理化所有集合类的结构。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，这些子类在某种程度上是规范的。哥德尔发现，存在着足够丰富的子类来满足ZFC公理，但又足够约束，使得它们的所有属性都可以被确定。这样的子类称为规范内部模型。例如，在规范内部模型中，连续统假设是正确的。通过考虑正则内模型中成立的陈述，得到ZFC的扩展。一个这样的陈述是超级力量公理，由PI在他的论文中确定和研究。公理强加了一个丰富的结构上的层次的上游无限的基数，或大基数，一个中心的研究对象，在集理论。利用Woodin最近的一项突破，PI能够对Ultrapower公理进行强有力的概括，该项目建议进行研究，希望它们能够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NSF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。