Descriptive Inner Model Theory and Its Applications

Zermelo-Fraenkel公理加上选择公理（Axiom of Choice， ZFC）已经被广泛接受为数学的基础；几乎所有被研究和应用于其他科学领域的数学分支都可以在ZFC中得到解释。然而，事实证明，有一些自然的和重要的数学理论是不能由ZFC单独决定的。大基数公理（lca）是ZFC的扩展，旨在解决所有这些理论。这是集合论中的哥德尔程序。如果LCA是“正确的”，那么它所决定的理论也是正确的。如何测试LCA的正确性？现代集合论中的一个主要项目“内模型计划”通过为lca构建规范模型来证明其正确性，就像自然数是皮亚诺算术公理（PA）的规范模型一样（因此PA是一个正确的理论）。该项目通过从ZFC的各种扩展中提出构建lca规范模型的方法，为内部模型计划做出了贡献。该项目重点研究内部模型、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（如适当强迫公理（PFA））之间的联系。PI提出将致力于推进混合结构的基础理论，重点关注策略小鼠和方法小鼠，并开发超越其现状的核心模型归纳方法。特别是，该项目旨在解决描述性内模型理论中的以下基本问题：(1)确定性模型的HOD是否具有精细结构（例如广义连续介质假设（GCH），各种平方原理在HOD中是否成立）？(2) PFA的稠度是多少？(3) PFA（或任何其他强组合理论）是否意味着各种部分背景结构中的模型是可迭代的？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。

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.