Combinatorial Set Theory

研究者提出了一个公理化集合论的研究计划。该程序涉及到许多不同的问题，所有这些问题都涉及到组合集理论，并且与大基数和强迫有关；要研究的领域包括相互平稳性、大基数的同一性危机和组合原则。使用的技术包括Radin强迫、迭代强迫和PCF理论。组合集理论是一门学科，其目标是采用有限集的熟悉概念（如计数，排序和置换）并将其扩展到无限集的环境中。该理论得到了高度发展，并在使用无限集的几个数学领域中得到了应用，包括拓扑和分析。研究者提出的问题的进展将增加我们对无限集的理解，并有助于形成对集合理论家和其他领域工作者有用的工具。

The investigator proposes a program of research in axiomatic set theory. The program involves looking at a number of different questions, all of which involve combinatorial set theory and are related to large cardinals and forcing; among the areas to be studied are mutual stationarity, identity crises for large cardinals, and combinatorial principles. The techniques to be used include Radin forcing, iterated forcing and PCF theory. Combinatorial set theory is a discipline whose goal is to take familiar ideas about finite sets (such as counting, ordering and permuting) and extend them to the context of infinite sets. The theory is highly developed and has found applications in several mathematical areas where infinite sets are used, including topology and analysis. Progress on the problems which are proposed by the investigator should increase our understanding of infinite sets, and help forge tools which will be useful to set theorists and workers in other areas.