Descriptive Set Theory and Categorical Logic

这个项目将集中在数理逻辑的两个不同方面之间的相互作用：分类逻辑和描述集合论。 从广义上讲，数理逻辑是对数学语言及其意义的形式化研究。 例如，断言“每两个数字之间，有第三个”的含义取决于“数字”（整数，分数等）的环境宇宙。一个是指。 范畴逻辑提供了构建“普遍抽象宇宙”的工具，在这个宇宙中，对数学陈述的解释同时在所有可能的“真实的”宇宙中捕捉它们的含义。 描述性集合论提供了对所有可能的“真实的”宇宙进行分类的工具，或者在某些情况下，证明不存在这样的分类。 最近的工作已经指向了这两个历史上完全不同的逻辑子领域之间的深刻联系，这个项目旨在进一步发展这种联系。在这个项目中，PI研究范畴逻辑和描述集合论作为无穷命题和一阶逻辑的双重“句法”和“语义”表示。 在命题设置中，一长串的工作指向locale理论（“无点拓扑”）作为波兰和标准Borel空间中经典描述集理论的不可数推广，并且该项目旨在通过开发更高级经典技术的localic对应物来扩展这种类比，例如投影集，有效描述集理论和强迫。 在一阶设置中，PI最近的工作表明，Toposes的Joyal-Tierney表示定理产生了无穷一阶逻辑理论与其可数模型的标准Borel群胚之间的对偶。 该项目旨在将这种二重性扩展到度量结构的连续逻辑，这将首先涉及开发拓扑理论的度量模拟的基础。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This project will focus on the interaction between two different aspects of mathematical logic: categorical logic and descriptive set theory. Broadly speaking, mathematical logic is the formal study of mathematical language and its meaning. For example, the assertion "between every two numbers, there is a third" changes meaning depending on which ambient universe of "numbers" (integers, fractions, etc.) one is referring to. Categorical logic provides tools for constructing a "universal abstract universe" in which the interpretation of mathematical statements captures their meanings in all possible "real" universes at once. Descriptive set theory provides tools for classifying all such possible "real" universes, or in certain cases, proving that no such classification exists. Recent work has pointed toward a deep connection between these two historically disparate subfields of logic, and this project aims to further develop this connection.In this project the PI studies categorical logic and descriptive set theory as dual "syntactic" and "semantic" representations of infinitary propositional and first-order logic. In the propositional setting, a long line of work points to locale theory ("point-free topology") as an uncountable generalization of classical descriptive set theory in Polish and standard Borel spaces, and this project aims to extend this analogy by developing localic counterparts of more advanced classical techniques, such as projective sets, effective descriptive set theory, and forcing. In the first-order setting, recent work by the PI has shown that the Joyal-Tierney representation theorem for toposes yields a duality between theories in infinitary first-order logic and their standard Borel groupoids of countable models. This project aims to extend this duality to continuous logic for metric structures, which will involve first developing the foundations for a metric analog of topos theory.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.