Set Theory

集合论

• 批准号: 1460238
• 负责人: William Woodin
• 金额: $ 30.55万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1460238&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。传统的观点认为，这一计划一直是一个增量计划的进展衡量的进展了层次结构的大枢机主教。现在知道这种看法是不正确的。一个临界转变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型问题时，人们解决了基本上所有已知的大基数公理的内模型问题。这个解必然产生哥德尔内部模型L的一个最终的扩大。这个终极-L必须紧密地近似于它所构造的母宇宙V。虽然Ultimate-L的详细构造是开放的，但公理“V = Ultimate-L”可以精确地表述，甚至现在也可以探索其后果。人们也可以精确地指定一个猜想，这将是Ultimate-L构造的最终结果。这就是终极L猜想。这个猜想与其他几个猜想密切相关，这些都是本研究计划的重点。无穷大的数学研究是集合论的重点。这个问题开始的基础上的原则孤立康托，然后阐述了其他人。到了世纪早期，基本的策梅洛-弗兰克尔公理被分离出来，这些公理与选择公理一起成为ZFC公理。ZFC公理定义了（数学）无限的当前概念。世纪后半叶的开创性发现表明，大多数关于无限集合的基本问题都无法在ZFC公理的基础上解决。著名的这些无法解决的问题是康托的连续统假设，其中中期的20世纪被广泛认为是最重要的问题之一的数学。 该项目的重点是Ultimate-L项目。这个项目如果成功将隔离（第一次）ZFC公理的扩展，将基本上解决所有这些否则无法解决的问题（包括连续统假设），并提供一个框架，将集合论的所有问题减少到大无限集的存在问题-这些就是所谓的强无穷公理。 新的方法产生于一系列最近的结果表明，这样的ZFC公理的扩展可能实际上存在。