Mathematical Sciences: Set Theory and Large Cardinals

这个项目将继续研究集合论的几个主题，主要是关于核心模型及其应用。这项研究最重要的目的将是扩展核心模型理论，以便人们可以在伍丁红衣主教在场的情况下证明核心模型的存在。这个核心模型的可能结构已经是已知的；主要问题似乎是开发可以用来证明扩展器是核心模型的成员的技术。其他研究领域涉及核心模型在假设不存在具有Woodin基数的模型的情况下的应用。这些潜在的应用包括与绝对性有关的问题，与奇异基数假设有关的问题，以及与琼森和拉姆齐基数有关的问题。这项研究的最终目的是刻画集合论的良好基础模型的结构。顾名思义，集合论宇宙的“核心”，如果它存在的话，它提供了一个具有相对定义良好的结构的骨架，从而提供了关于完整模型形式的大量信息。由于集合论是数学的基础，这项事业对许多公认的数学实践的可靠性有着重要的影响，尽管大多数工作的数学家习惯于相信这一点。

This project will continue to study several topics in set theory, primarily concerning the core model and its applications. The most important aim of this research will be to extend core model theory so that one can prove the existence of a core model in the presence of a Woodin cardinal. The probable structure of this core model is already known; the main problem seems to be to develop techniques which can be used to prove an extender is a member of the core model. Other areas of study concern applications of the core model under the assumption that there is no model with a Woodin cardinal. These potential applications include questions related to absoluteness, to the singular cardinal hypothesis, and to Jonson and Ramsey cardinals. The ultimate aim of this research is to chararcterize the structure of well-founded models of set theory. As their name suggests, the 'core' of a universe of set theory, if it exists, provides a skeleton with a relatively well-defined structure and thus gives a good deal of information about the form of the full model. Since set theory lies at the very foundation of mathematics, this enterprise has important consequences for the soundness of many accepted mathematical practices, although it is customary for most working mathematicians to take this on faith.