CAREER: Current and Future Developments of the Core Model Induction

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Gödel在集合论中的程序。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，为内部模型程序做出了贡献。教育部分包括年度会议、课程发展、外联和扩大参与活动。该项目符合研究内部模型、实数集、混合结构（如确定性模型的遗传序数可定义集（HOD））、强迫和强组合原理（如适当强迫公理（PFA））之间联系的一般框架。为LCA构建规范的内部模型的问题称为该LCA的内部模型问题。该项目的主要目标之一是确定可能阻碍当前解决各种lca内部模型问题的方法进一步发展的某些障碍，并提出克服这些障碍的方法。具体而言，该项目将研究当前核心模型归纳法技术的障碍（其中一个障碍是普遍贝尔集的密封），并开发克服这些障碍的核心模型归纳法的新方法。上述分析的一个主要应用是从ZFC的各种强扩展（如适当强迫公理）构造大基数的规范内模型。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 Gödel'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). This project contributes to the Inner Model Program by advancing methods for constructing canonical models for LCAs from various extensions of ZFC. The educational component includes annual conferences, curriculum development, and outreach and broadening participation activities. The project fits into the general framework of 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 problem of building a canonical inner model for an LCA is referred to as the Inner Model Problem for that LCA. One of the project's main goals is to identify certain obstructions that may prevent the current approaches to resolving the Inner Model Problem for various LCAs from going further as well as proposing ways of overcoming them. Specifically, the project will study obstacles to the current techniques of the Core Model Induction (one such obstacle is the Sealing of the Universally Baire Sets) and develop new approaches to the Core Model Induction that overcome these obstacles. One main application of the above analyses is in constructing canonical inner models of large cardinals from various strong extensions of ZFC, like the Proper Forcing Axiom.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.