The HOD Project

集合论是数学领域，主要研究无穷大的数学问题。基本问题包括康托的连续统假说。这是希尔伯特在1900年提出的现在著名的清单上的第一个问题。根据Godel在1940年和Cohen在1963年的结果，根据集合论的公理，这个问题已经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年中用来证明的所有其他问题也是不可解的。这一新公理与大型基数公理的兼容性是现在的中心问题。这个项目的重点是通过证明这一新公理不能被大型基数公理驳斥，或者通过证明一些合理的大型基数公理驳斥了这一新公理，来得出对这一新公理的分析结论。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L猜想提出的根本问题。这个定理表明，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猜想问题的唯一可能途径是L猜想。这个项目的重点是两个部分。或者把内模推广到超紧基数的水平来证明L猜想，这就证明了L猜想，或者驳斥了L猜想，因为L猜想是错误的。后一种可能性现在是一个合理的研究目标，因为越来越多的约束被发现，超紧基数的内模理论必须满足这些约束。无论如何，这个项目的成功结束将解决公理``V=终极‘与大型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。这一裁决反映了NSF的法定使命，并通过使用基金会的智力优势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.