The HOD Project

HOD项目

• 批准号: 1953093
• 负责人: William Woodin
• 金额: $ 30万
• 依托单位: Harvard University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2020
• 资助国家: 美国
• 起止时间: 2020-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1953093&HistoricalAwards=false
• 关键词: HOD; Project

Set Theory is the area of Mathematics which is focused on the mathematical study of infinity. The fundamental questions include that of Cantor’s Continuum Hypothesis. This was the first question on the now famous list of proposed by Hilbert in 1900. By the results of Godel in 1940 and Cohen in 1963, this problem has been shown to be formally unsolvable on the basis of the axioms of Set Theory. But this does not mean that the question has no answer, rather it simply shows the accepted axioms of Set Theory are incomplete. In previously funded research, a single new axiom has been discovered which resolves not only the problem of the Continuum Hypothesis but essentially all the other questions which Cohen’ method has been used in the last 50 years to show are also unsolvable. The compatibility of this new axiom with large cardinal axioms is now the central problem. The focus of this project is to bring the analysis of this new axiom to a conclusion, by either showing this new axiom cannot be refuted by large cardinal axioms, or by showing that some reasonable large cardinal axiom refutes this new axiom. In addition the project also provides research training opportunities for graduate students.The HOD Dichotomy Theorem isolates the fundamental issue raised by the Ultimate L Conjecture. This theorem shows that assuming reasonable large cardinal axioms, HOD must be either very close to V or very far from V. The HOD Conjecture is the conjecture that the “far option” is vacuous. The only plausible approach at present to resolving the problem of the HOD Conjecture lies in the Ultimate L Conjecture. The focus of this project is in two parts. Either prove the Ultimate L Conjecture by extending inner model to the level of a supercompact cardinal, this would prove the HOD Conjecture, or refute the Ultimate L Conjecture which would strongly argue that the HOD Conjecture is false. The latter possibility is now a reasonable research target because of the increasing number of constraints which have been discovered that the inner model theory for a supercompact cardinal must satisfy. Either way, the successful conclusion of this project will resolve the problem of the compatibility of the axiom ``V=Ultimate” with large cardinal axioms. Depending on the outcome, one either verifies the HOD Conjecture or discovers why the HOD Conjecture is likely false.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.

集合论是数学领域，专注于无穷大的数学研究。基本问题包括康托连续统假设的问题。这是希尔伯特在 1900 年提出的现在著名的问题列表中的第一个问题。哥德尔在 1940 年和科恩在 1963 年的结果表明，这个问题在集合论公理的基础上在形式上是无法解决的。但这并不意味着这个问题没有答案，而只是表明集合论公认的公理是不完整的。在之前资助的研究中，发现了一个新公理，它不仅解决了连续统假说的问题，而且基本上解决了过去 50 年来科恩方法所显示的所有其他问题也是无法解决的。这个新公理与大基本公理的兼容性现在是中心问题。该项目的重点是通过证明这个新公理不能被大基数公理反驳，或者通过证明一些合理的大基数公理反驳这个新公理，从而对这个新公理的分析得出结论。此外，该项目还为研究生提供研究培训机会。HOD二分定理隔离了终极L猜想提出的基本问题。该定理表明，假设合理的大基数公理，HOD 必须要么非常接近 V，要么非常远离 V。 HOD 猜想是“远选项”是空的猜想。目前解决HOD猜想问题唯一可行的方法是终极L猜想。该项目的重点分为两个部分。要么通过将内模型扩展到超紧基数的水平来证明终极 L 猜想，这将证明 HOD 猜想，要么反驳终极 L 猜想，这将有力地证明 HOD 猜想是错误的。后一种可能性现在是一个合理的研究目标，因为已经发现超紧基数的内模型理论必须满足的约束数量不断增加。不管怎样，这个项目的成功结束将解决公理“V=终极”与大基数公理的兼容性问题。根据结果，人们要么验证 HOD 猜想，要么发现为什么 HOD 猜想可能是错误的。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。