Forcing with Large Cardinals

我们的数学学习是通过严格的公理演绎，即证明来进行的。众所周知，这项研究的结果是不完整的，可能会导致无法根据数学基本原理(ZFC公理)决定的陈述。集合论是一个数理逻辑领域，它为分析数学语句的一致性提供了工具。获得一致性结果的系统方法是通过构建可在其中检查和验证各种陈述的一致性的替代数学宇宙来实现的。这个项目涉及几种这样的施工方法。它的目的是开发新的方法来解决无限组合数学中的几个重要问题，并研究现有的方法及其局限性。主要研究对象是(I)集合论中的代数；(Ii)钻石序列；(Iii)Mitchell阶；(Iv)遗传有序可定义集(HOD)的内部模型。关于(I)，该项目旨在研究由Foreman和Magidor引入的不同的奇异平稳概念，并在回答关于Jonsson代数的问题方面取得进展。关于(2)，该项目讨论了钻石类型原则及其与紧凑性原则和基本算术假设的相互作用。关于(Iii)，本项目旨在通过研究各种Mitchell序结构的一致性强度问题来研究强迫理论和内模理论之间的联系。关于(Iv)，该项目旨在研究HOD与集合论宇宙V的接近程度，并解决HOD猜想。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。

Our study of mathematics is carried out by means of rigorous deduction from axioms, i.e., proofs. It is well-known that the results of this study are incomplete, and may lead to statements which cannot be decided from the fundamental principles of mathematics (the ZFC axioms). Set theory is a field of mathematical logic which provides tools to analyze the consistency of mathematical statements. The systematic approach by which consistency results are obtained comes through the construction of alternative mathematical universes in which the consistency of various statements can be examined and validated. This project deals with several such construction methods. It aims to develop new methods to address several important problems in infinitary combinatorics, and to study existing methods and their limitations. The main objects of study are (i) algebras in set theory; (ii) diamond sequences, (iii) the Mitchell order; and (iv) inner models of hereditarily ordinal definable sets (HOD). In connection with (i), the project aims to study different notions of singular stationary, which were introduced by Foreman and Magidor, and make advancements in answering questions regarding Jonsson algebras. In connection with (ii), the project addresses diamond type principles and their interaction with compactness principles and cardinal arithmetic assumptions. In connection with (iii), the project aims to study the connection between forcing theory and inner model theory by studying problems concerning the consistency strength of various Mitchell order structures. In connection with (iv), the project aims to study the extent to which HOD is close to the set-theoretic universe V, and address the HOD-conjecture.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.