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在ZFC内部决定。大型基数公理(LCA)是ZFC的扩展，旨在解决所有这些理论。因此，LCA在集合论中追求戈德尔的计划。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是一个正确的理论)。这项研究项目通过提出从ZFC的各种扩展构建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)、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本理论，以及超越其现状的核心模型归纳的发展方法。特别是，该项目的目的是在回答描述性内模型理论中的两个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，广义连续统假设(GCH)和各种平方原理在HOD中成立吗)？(2)PFA的一致性强度是什么？

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?