Combinatorics, Large Cardinals and Forcing

将承担的项目是集合论的研究计划，调查公理集合论Zermelo-Fraenkel with Choice(ZFC)。要研究的课题包括奇异基数的组合学，关于Souslin树和Aronszajn树的问题，Woodin的一般迭代理论和“正则”模型，以及Radin强迫的理论和应用。与伍丁合著的一本关于雷丁强迫及其应用的说明性书籍将完成。程序中使用的技术有组合集合论、强迫、内模论和描述集合论。正在考虑的问题的统一主题是，它们都以这样或那样的方式与大型基数联系在一起。该方案的目的是利用数理逻辑中的工具来研究关于无限集的一些数学问题。在集合论中，所有的数学对象都可以用集合来表示，所有的数学证明都可以用证明来表示，无限集合论是现代数学的核心。数理逻辑是一门应用数学技术来证明有关“形式系统”的结果的学科，如公理数学理论或(最近)计算机编程语言。在这些问题上的进步应该导致新的数学和新的逻辑工具的发展。

The project to be undertaken is a program of research in set theory, investigating the axiomatic set theory Zermelo-Fraenkel with Choice (ZFC). The topics to be investigated include the combinatorics of singular cardinals, questions about Souslin and Aronszajn trees, Woodin's theory of genepic iterations and "canonical" models, and the theory and applications of Radin forcing. An expository book with Woodin about Radin forcing and its applications will be completed. The techniques to be used in the program are those of combinatorial set theory, forcing, inner model theory and descriptive set theory. The unifying theme of the problems under consideration is that they are all connected in some way or other with large cardinals. The aim of this proposal is to investigate some mathematical problems about infinite sets, using tools from mathematical logic. The theory of infinite sets lies at the heart of modern mathematics, in the sense that all mathematical objects can be represented as sets and all mathematical proofs can be represented as proofs in set theory. Mathematical logic is a discipline in which mathematical techniques are applied to prove results about "formal systems" such as axiomatic mathematical theories or (more recently) computer programming languages. Progress on the problems should lead to new mathematics and the development of new logical tools.