The HOD Project

集合论是数学的一个领域，它关注于无穷的数学研究。基本问题包括康托的连续统假设。这是希尔伯特1900年提出的著名问题清单中的第一个问题。哥德尔（1940年）和科恩（1963年）的结果表明，在集合论公理的基础上，这个问题在形式上是不可解的。但这并不意味着这个问题没有答案，它只是表明了公认的集合论公理是不完整的。在先前资助的研究中，发现了一个新的公理，它不仅解决了连续统假设的问题，而且基本上解决了科恩方法在过去50年里用来证明也是无法解决的所有其他问题。这个新公理与大基数公理的相容性现在是中心问题。本课题的重点是对这个新公理的分析得出结论，要么表明这个新公理不能被大基数公理所反驳，要么表明一些合理的大基数公理可以反驳这个新公理。此外，该项目还为研究生提供研究培训机会。HOD二分定理孤立了终极L猜想提出的基本问题。这个定理表明，假设合理的大基本公理，HOD必须非常接近V或非常远离V。HOD猜想是关于“远选项”是空的猜想。目前解决HOD猜想问题的唯一可行的方法是终极L猜想。本项目的重点分为两部分。要么通过将内部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，这将强烈地证明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，因为已经发现了越来越多的约束条件，这些约束条件是超紧基数的内模理论必须满足的。无论如何，这个项目的成功结论将解决公理“V=Ultimate”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。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.