Forcing with Large Cardinals

我们对数学的研究是通过从公理中进行严格的演绎来进行的，即，证明。众所周知，这项研究的结果是不完整的，并可能导致无法从数学的基本原则（ZFC公理）决定的陈述。集合论是数理逻辑的一个领域，它提供了分析数学陈述的一致性的工具。系统的方法，其中获得的一致性结果来自通过建设替代数学宇宙中的各种声明的一致性可以检查和验证。本项目涉及几种这样的施工方法。它旨在开发新的方法来解决无穷组合学中的几个重要问题，并研究现有的方法及其局限性。主要的研究对象是（i）集合论中的代数;（ii）菱形序列;（iii）米切尔序;（iv）遗传序可定义集（HOD）的内部模型。关于（i），该项目旨在研究由Foreman和Magidor引入的奇异平稳的不同概念，并在回答有关Jonsson代数的问题方面取得进展。关于（二），该项目讨论了钻石型原则及其与紧凑性原则和基数算术假设的相互作用。 关于（iii），本课题通过研究各种Mitchell序结构的一致性强度问题，研究强迫理论与内模理论之间的关联。关于（iv），该项目旨在研究HOD接近集合论宇宙V的程度，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。

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.