Descriptive Inner Model Theory and Its Applications

Zermelo-Fraenkel公理加上选择公理(ZFC)已被广泛接受为数学的基础；几乎所有已被研究并应用于其他科学领域的数学分支都可以用ZFC来解释。然而，事实证明，有一些自然而重要的数学理论不能仅由ZFC来决定。大型基数公理(LCA)是ZFC的扩展，旨在解决所有这些理论。这是戈德尔在集合论中的程序。如果生命周期评价是“正确的”，那么它决定的理论也是正确的。如何测试生命周期评价的正确性？内模程序是现代集合论中的一个主要程序，它通过构建LCA的规范模型来证明正确性，就像自然数是Peano算术公理(PA)的规范模型一样(同样，PA也是一个正确的理论)。拟议的项目通过提出从ZFC的各种扩展中构建LCA的规范模型的方法来为内部模型计划做出贡献。该项目专注于研究内部模型、实数集、混合结构(如确定性模型的遗传有序可定义集(HOD))、强迫和强组合原理(如真强迫公理(PFA))之间的联系。PI建议致力于推进杂交结构的基本理论，重点关注策略小鼠和hod小鼠，以及开发超越其当前状态的核心模型诱导方法。特别是，该项目的目的是在回答描述性内模型理论中的以下基本问题方面取得进展：(1)确定性模型的HOD是否是精细结构的(例如，广义连续假设(GCH)、各种平方原理在HOD中是否成立)？(2)PFA的一致性强度是什么？(3)PFA(或任何其他强组合理论)是否意味着各种部分有背景的结构中的模型是可迭代的？该奖项反映了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。

The Zermelo-Fraenkel axioms plus the Axiom of Choice (ZFC) have been widely accepted as a foundation for mathematics; virtually all branches of mathematics that have been studied and applied to other scientific fields can be interpreted in ZFC. However, it turns out that there are natural and important mathematical theories which cannot be decided by ZFC alone. The Large Cardinal Axioms (LCAs) are extensions of ZFC designed to settle all such theories. This is the Godel's program in Set Theory. If an LCA is "correct", then the theories it decides are also correct. How can one test for correctness of an LCA? The Inner Model Program, a major program in modern Set Theory, justifies correctness by constructing canonical models for LCAs much like the natural numbers are the canonical model for the Peano Axioms of Arithmetic (PA) (and as such PA is a correct theory). The proposed project contributes to the Inner Model Program by advancing methods for constructing canonical models for LCAs from various extensions of ZFC. The project focuses on studying the connections between inner models, sets of reals, hybrid structures (such as hereditarily ordinal definable sets (HOD) of determinacy models), forcing, and strong combinatorial principles (such as the Proper Forcing Axiom (PFA)). The PI proposes to work on advancing the basic theory of hybrid structures, focusing on strategy mice and hod mice as well as developing methods for the core model induction beyond its current state. In particular, the project aims to make advancements in answering the following fundamental questions in descriptive inner model theory: (1) Is HOD of a determinacy model fine-structural (e.g. do the Generalized Continuum Hypothesis (GCH), various square principles hold in HOD)? (2) What is the consistency strength of PFA? (3) Does PFA (or any other strong combinatorial theory) imply models in various partially backgrounded constructions iterable?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.