Set Theory

集合论

• 批准号: 1301658
• 负责人: William Woodin
• 金额: $ 48万
• 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: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2014-10-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1301658&HistoricalAwards=false
• 关键词: Set; Theory

The Inner Model Program is one of the central areas of Set Theory. The traditional view of this program has been that of an increment program with progress measured by a progression up the hierarchy of large cardinals. It is now known that this view is not correct. A critical transition occurs at the level of exactly one supercompact cardinal, and in solving the Inner Model Problem for this specific large cardinal axiom, one solves the Inner Model Problem for essentially all known large cardinal axioms. The solution must necessarily yield an ultimate enlargement of Godel's inner model L. This Ultimate-L must closely approximate the parent universe V within which it is constructed. Though the detailed construction of Ultimate-L is open, the axiom, ``V = Ultimate-L'', can be precisely formulated and its consequences explored even now. One can also precisely specify a conjecture which would be the final outcome of the construction of Ultimate-L. This is the Ultimate-L Conjecture. This conjecture is closely related to several other conjectures and these collectively are the focus of this research proposal.The mathematical study of Infinity is the focus of Set Theory. This subject began on the basis of principles isolated by Cantor and then elaborated on by others. By the early part of the 20th century the basic Zermelo-Frankel Axioms had been isolated and these axioms together with the Axiom of Choice are the ZFC axioms. The ZFC axioms define the current conception of (mathematical) Infinity. The seminal discoveries of the latter half of the 20th century showed that most of the fundamental questions asked about infinite sets are unsolvable on the basis of the ZFC axioms. Famous among these unsolvable questions is that of Cantor's Continuum Hypothesis which by the middle of the 20th century was widely regarded as one of the most important questions of mathematics. This proposal focuses on the Ultimate-L Project. This project if successful will isolate (for the first time) an extension of the ZFC axioms that will resolve essentially all of these otherwise unsolvable questions (including that of the Continuum Hypothesis) and provide a framework for reducing all questions of Set Theory to questions of the existence of large infinite sets--these are the so called Axioms of Strong Infinity. The new approach arises from a series of recent results which show that such an extension of the ZFC axioms might actually exist.

内模程序是集合论的核心领域之一。这个程序的传统观点是一个增量程序，其进度是通过大基数的层次结构的进展来衡量的。现在知道这种观点是不正确的。一个临界跃迁恰好发生在一个超紧基数的水平上，在解决这个特定的大基数公理的内模问题时，基本上解决了所有已知的大基数公理的内模问题。这个解必须产生哥德尔内模型l的最终扩展。这个最终模型l必须非常接近于它所在的母宇宙V。虽然最终- l的详细结构是开放的，公理，“V =最终- l”，可以精确地表述和它的后果探索，甚至现在。人们也可以精确地指定一个猜想，它将是最终l构造的最终结果。这就是终极l猜想。这一猜想与其他几个猜想密切相关，这些猜想共同成为本研究计划的重点。无穷的数学研究是集合论的重点。这一课题以康托尔孤立的原则为基础，然后由其他人加以阐述。到20世纪早期，基本的泽梅洛-弗兰克尔公理已经被分离出来，这些公理与选择公理一起被称为ZFC公理。ZFC公理定义了（数学）无限的当前概念。20世纪下半叶的开创性发现表明，大多数关于无限集的基本问题在ZFC公理的基础上是无法解决的。在这些无法解决的问题中，著名的是康托的连续统假设，它在20世纪中叶被广泛认为是数学中最重要的问题之一。本提案的重点是Ultimate-L项目。这个项目如果成功，将（第一次）分离出ZFC公理的扩展，它将基本上解决所有这些无法解决的问题（包括连续统假设的问题），并提供一个框架，将集合论的所有问题简化为大无穷集的存在性问题——这些就是所谓的强无穷公理。这种新方法源于最近的一系列结果，这些结果表明ZFC公理的这种扩展可能确实存在。