Large cardinals and small sets

大红衣主教和小红衣主教

• 批准号: 1201494
• 负责人: Paul Larson
• 金额: $ 12.24万
• 依托单位: Miami University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-07-15 至 2018-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1201494&HistoricalAwards=false
• 关键词: Large; cardinals; small; sets

Larson proposes work on five related topics, all centering around the extent to which the absoluteness results and detailed analysis of inner models of determinacy in the context of large cardinals can be lifted to larger models satisfying larger fragments of the Axiom of Choice. One topic is the so-called universally measurable sets, where the fundamental open question is whether all such sets can have the property of Baire. One topic concerns the fragment of the Axiom of Choice which holds when a model of determinacy is expanded by adding a Ramsey ultrafilter. Another concerns lifting to the third uncountable cardinal Woodin's results on getting forcing axioms at the second uncountable cardinal by forcing over a determinacy model. The fourth concerns application of techniques from set theory to Shelah's study of Abstract Elementary Classes. Finally, Larson proposes to study ideals on the first uncountable cardinal from the point of view of certain inner models satisfying a small fragment of Choice. The standard axioms for set theory, the Zermelo-Fraenkel axioms, serve as the commonly accepted foundations for mathematics, though as mathematics becomes more abstract, more and more issues arise which are not resolved by these axioms. Many of these issues are studied by set theorists, in hope of finding the right extension of these axioms. Developments in the study of such possible extensions have had a dramatic foundational impact in the last thirty years, affecting many areas of mathematics, and even philosophy. The PI works on the border between some of the more technical, inward-directed areas of set theory and more classical areas with connections to other fields. Much of his work consists of finding applications of these more technical areas, and exposing them to a wider audience.

拉森提出了五个相关主题的工作，所有主题都围绕着大基数背景下确定性内部模型的绝对性结果和详细分析可以提升到满足选择公理更大片段的更大模型的程度。其中一个主题是所谓的普遍可测集，其中基本的悬而未决的问题是是否所有此类集都可以具有拜尔性质。其中一个主题涉及选择公理的片段，当通过添加拉姆齐超滤器来扩展确定性模型时，该公理成立。另一个问题涉及将伍丁的结果提升到第三个不可数基数，通过强制确定性模型在第二个不可数基数上获得强制公理。第四个涉及集合论技术在 Shelah 对抽象基本类的研究中的应用。最后，拉尔森建议从满足选择的一小部分的某些内部模型的角度来研究第一个不可数基数上的理想。集合论的标准公理，即策梅洛-弗兰克尔公理，是普遍接受的数学基础，尽管随着数学变得越来越抽象，出现了越来越多的问题，而这些公理无法解决。集合论学家研究了其中许多问题，希望找到这些公理的正确扩展。在过去的三十年里，对这种可能的扩展的研究的发展产生了巨大的基础性影响，影响了数学甚至哲学的许多领域。 PI 的工作介于集合论的一些技术性更强、内向性领域和与其他领域有联系的经典领域之间。他的大部分工作包括寻找这些技术性更强的领域的应用，并将其展示给更广泛的受众。