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.

内模计划是集合论的核心领域之一。这个方案的传统观点一直是一个增量方案，其进展是通过大基数等级的递增来衡量的。现在已经知道，这种观点是不正确的。一个关键的转变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，基本上解决了几乎所有已知的大基数公理的内模问题。这个解决方案必然会导致哥德尔内部模型L的最终放大。这个终极--L必须非常接近构成它的母宇宙V。虽然《终极L》的具体构造尚无定论，但《L终极》这一公理至今仍能得到精确的表述及其后果的探究。人们还可以精确地指定一个猜想，这将是最终结果的终极-L。这就是终极-L猜想。这一猜想与其他几个猜想密切相关，这些猜想都是本研究方案的重点。对无穷的数学研究是集合论的重点。这一主题开始于康托尔所孤立的原则，然后由其他人详细阐述。到20世纪初，基本的Zermelo-Frankel公理已经被孤立，这些公理和选择公理一起就是ZFC公理。ZFC公理定义了当前的(数学)无限的概念。20世纪后半叶的开创性发现表明，在ZFC公理的基础上，提出的关于无限集的大多数基本问题都是不可解的。在这些无法解决的问题中最著名的是康托的连续统假设，到20世纪中叶，它被广泛认为是最重要的数学问题之一。这份提案的重点是终极项目--L项目。这个项目如果成功，将隔离(第一次)ZFC公理的扩展，它将基本上解决所有这些原本无法解决的问题(包括连续统假设)，并提供一个框架，将集合论的所有问题归结为大型无限集的存在问题--这些就是所谓的强无穷大公理。这一新方法源于最近的一系列结果，这些结果表明ZFC公理的这种扩展实际上可能存在。