Large cardinals and small sets

拉尔森提出了五个相关主题的工作，都围绕着在多大程度上的绝对性结果和详细分析的内部模型的确定性的背景下，大基数可以提升到更大的模型，满足更大的片段的选择公理。一个主题是所谓的普遍可测集，其中基本的开放问题是是否所有这样的集合都可以具有Baire属性。一个主题涉及片段的公理选择举行时，一个模型的确定性是扩大了增加一个拉姆齐超滤。另一个问题是提升到第三个不可数基数Woodin的结果，在第二个不可数基数上通过强迫确定性模型得到强迫公理。第四个问题的应用技术从集合论希拉的研究抽象的小学类。最后，拉尔森建议研究理想的第一不可数基数的角度来看，某些内部模型满足一个小片段的选择。集合论的标准公理，即策梅洛-弗兰克尔公理，是数学普遍接受的基础，尽管随着数学变得越来越抽象，越来越多的问题出现，这些公理无法解决。集合理论家研究了其中的许多问题，希望找到这些公理的正确扩展。在过去的30年里，对这种可能的扩张的研究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工作在集合论的一些技术性更强、更内向的领域和与其他领域有联系的更经典的领域之间的边界上。他的大部分工作都是寻找这些技术性更强的领域的应用，并将它们展示给更广泛的受众。

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.