Descriptive Inner Model Theory, Large Cardinals, and Combinatorics

集合论的标准公理Zermelo-Fraenkel集合论和选择公理（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，有一些自然而重要的数学理论是不能单独在ZFC内决定的。大基数公理（lca）是ZFC的扩展，旨在解决所有这些理论。因此lca在集合论中追求哥德尔的程序。如何测试LCA的“正确性”？内部模型程序是现代集合理论中的一个主要程序，它通过为lca构建规范模型来证明正确性，就像自然数是皮亚诺算术公理（Peano Axioms of arithmetic， PA）的规范模型一样（因此PA是一个正确的理论）。本研究项目通过从ZFC的各种扩展中提出构建lca规范模型的方法，为内模型规划做出了贡献。该项目侧重于研究内部模型、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（如适当强迫公理（PFA）、树性质的（推广）、唯一分支假设（UBH））之间的联系。本研究项目旨在推进混合结构的基础理论，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两个基本问题：(1)确定性模型的HOD是否具有精细结构（例如广义连续体假设（GCH）和各种平方原理在HOD中是否成立）？(2) PFA的稠度是多少？

The standard axioms of set theory, Zermelo-Fraenkel set theory with 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 that cannot be decided within ZFC alone. The Large Cardinal Axioms (LCAs) are extensions of ZFC designed to settle all such theories. Thus LCAs pursue Godel's program in set theory. 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 research 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), (generalizations of) the tree property, the Unique Branch Hypothesis (UBH)). This research project aims to advance the basic theory of hybrid structures, as well as developing methods for the core model induction beyond its current state. In particular, the project aims to make advancements in answering two fundamental questions in descriptive inner model theory: (1) Is HOD of a determinacy model fine-structural (e.g. do the Generalized Continuum Hypothesis (GCH) and various square principles hold in HOD)? (2) What is the consistency strength of PFA?