Mathematical Sciences: Set Theory and Large Cardinals

本项目将继续研究几个主题， 理论，主要是关于核心模型及其应用。 这项研究的最重要的目的将是扩大核心 模型理论，以便证明核心模型的存在 当着一位伍丁红衣主教的面 可能的结构 这个核心模型是已知的;主要问题似乎是 开发可用于证明扩展程序是 核心模型的成员。 其他研究领域 核心模型的应用，假设有 没有一个模特有伍丁枢机主教这些潜在的应用 包括与绝对性、单数 红衣主教假说，以及琼森和拉姆齐红衣主教。 本研究的最终目的是为了表征 集合论的良基模型的结构。 正如其名称 这表明，集合论宇宙的“核心”，如果它存在的话， 提供了具有相对明确的结构的骨架， 因此，提供了大量关于完整形式的信息。 模型 因为集合论是 数学，这项事业有重要的后果， 许多公认的数学实践的合理性，虽然它是 大多数在职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.