Combinatorics, Large Cardinals and Forcing

该项目将进行的是一个方案的研究集理论，调查公理集理论Zermelo-Fraenkel与选择（ZFC）。 要调查的主题包括组合学的奇异基数，问题的Souslin和Aronszajn树，Woodin的理论genepic迭代和“典型”的模式，以及理论和应用的Radin迫使。 一本关于Radin强迫及其应用的与Woodin的临时书将完成。在程序中使用的技术是组合集合论，强迫，内模理论和描述集合论。正在考虑的问题的统一主题是，他们都以某种方式或其他与大枢机主教。 这个建议的目的是研究一些数学问题的无限集，使用工具，从数理逻辑。无限集合论是现代数学的核心，因为所有的数学对象都可以表示为集合，所有的数学证明都可以表示为集合论的证明。数理逻辑是一门应用数学技术来证明关于“形式系统”的结果的学科，例如公理数学理论或（最近）计算机编程语言。这些问题的进展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.