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.

集合理论是数学领域，其重点是无限的数学研究。基本问题包括Cantor的连续假设。这是希尔伯特（Hilbert）在1900年提出的现在著名清单上的第一个问题。根据1940年的戈德尔（Godel）和1963年的科恩（Cohen）的结果，根据集合理论的公理，该问题已被证明是正式无法解决的。但这并不意味着这个问题没有答案，而是仅仅表明了集合理论的公理是不完整的。在先前资助的研究中，已经发现了一个新的公理，它不仅可以解决连续性假设的问题，而且从本质上讲，在过去50年中使用了Cohen方法的所有其他问题也无法解决。现在，这种新公理与大型基本公理的兼容性现在是中心问题。该项目的重点是通过表明该新公理不能被大型的基本公理驳斥，或者通过表明某些合理的大型基本公理会反驳这一新公理，从而使对新公理的分析得出结论。此外，该项目还为研究生提供了研究培训机会。HOD二分法定理隔离了最终的猜想提出的基本问题。该定理表明，假设合理的大型基本公理，HOD必须非常接近V或离V很远。HOD猜想是“远方选项”是真空的猜想。目前，解决HOD猜想问题的唯一合理方法在于最终的猜想。该项目的重点分为两个部分。要么通过将内部模型扩展到超级紧密的基数水平来证明最终的猜想，这将证明hod的猜想，或者反驳最终的L猜想，这将强烈认为HOD猜想是错误的。后来的可能性现在是一个合理的研究目标，因为越来越多的限制因素发现，超级紧张的基数的内部模型理论必须满足。无论哪种方式，该项目的成功结论都将解决公理与大型基本公理的兼容性问题。根据结果​​的不同，人们要么验证HOD的猜想，要么发现为什么HOD猜想可能是错误的。该奖项反映了NSF的法定任务，并通过使用基金会的知识分子和更广泛的影响审查标准来通过评估来诚实地表示支持。