Absoluteness and Choice

拉森计划研究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。这类问题中的一个主题是关于产生一个选择模型的问题，该模型具有与确定性的内部模型相同的序数基数。另一个问题是关于包含实数、序数和整数上的一个超滤子的最小模型是否坍缩包含实数和序数的最小内模型中的任何非序数基数的问题，相关的问题涉及选择模型的强迫绝对性，不可数长度博弈的确定性，第一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。许多集合论学家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20年里，对这种可能的扩展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的工作范围介于集合论中一些技术性较强、内向的领域和与其他领域有联系的较经典领域之间。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 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.