The HOD Project

集合论是数学领域，专注于无穷大的数学研究。基本问题包括康托的连续统假设。这是希尔伯特在1900年提出的第一个问题。根据哥德尔在1940年和科恩在1963年的结果，这个问题在集合论公理的基础上被证明是形式上不可解的。但这并不意味着这个问题没有答案，而只是表明集合论的公认公理是不完整的。在以前资助的研究中，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年中使用的所有其他问题，这些问题也是无法解决的。这个新公理与大基数公理的兼容性现在是中心问题。这个项目的重点是把这个新公理的分析得出结论，要么显示这个新公理不能被大基数公理反驳，或者显示一些合理的大基数公理反驳这个新公理。此外，该项目还为研究生提供了研究培训的机会。HOD二分法定理隔离了终极L猜想提出的基本问题。这个定理表明，假设合理的大基数公理，HOD必须非常接近V或非常远离V。HOD猜想是“远选项”是空的猜想。目前解决HOD猜想问题的唯一可行方法是终极L猜想。该项目的重点是两个部分。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，因为越来越多的限制，已被发现，内部模型理论的超紧基数必须满足。无论哪种方式，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。这个奖项反映了NSF的法定使命，并已被认为是值得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的支持。

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.