Absoluteness and Choice

拉森计划研究在多大程度上，在大基数的背景下，对确定性的绝对结果和内部模型的详细分析可以提升到满足选择公理的更大片段的更大模型。这类问题中的一个主题涉及到产生一个具有与某些内在确定性模型相同的序数基数的选择模型的问题。另一个问题是包含实数、序数和关于整数的一个超滤子的极小模型是否折叠包含实数和序数的最小内模型中的任何非序数基数。相关问题涉及选择模型的强制绝对性、无数长度对策的确定性以及第一个不可数基数上的非平稳理想的性质。集合论为数学提供了基础，因为集合论的公理旨在描述数学概念的宇宙。许多集合论者研究集合论的标准公理不能解决的问题，希望找到这些公理的正确扩展。在过去的二十年里，对这种可能的扩展的研究的发展已经产生了戏剧性的基础性影响，影响了数学的许多领域，甚至哲学。PI工作在集合论中一些更技术性、内向型领域和与其他领域有联系的更经典领域之间的边界上。他的大部分工作包括寻找这些更具技术性的领域的应用程序，并将它们展示给更广泛的受众。

Larson plans to study 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 in this class of problems concerns the problem of producing a model of Choice with the same ordinal cardinalities as some inner model of determinacy. Another concerns the question of whether the minimal model containing the reals, the ordinals and one ultrafilter on the the integers collapses any non-ordinal cardinals in the minimal inner model containing the reals and the ordinals.Related problems concern forcing-absoluteness for models of Choice, determinacy for games of uncountable length, and properties of the nonstationary ideal on the first uncountable cardinal.Set theory provides a foundation for mathematics, in that the axioms of set theory aim to describe the universe of mathematical concepts. Many set theorists study questions which are not resolved by the standard axioms for set theory, 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 twenty 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.