Descriptive Set Theory and Categorical Logic

这个项目将集中于数学逻辑的两个不同方面：范畴逻辑和描述集合论之间的相互作用。从广义上讲，数学逻辑是对数学语言及其意义的形式化研究。例如，断言“每两个数字之间存在第三个数字”的含义取决于所引用的“数字”（整数、分数等）的环境范围。范畴逻辑为构建一个“普遍的抽象宇宙”提供了工具，在这个宇宙中，对数学命题的解释可以同时捕捉到它们在所有可能的“真实”宇宙中的意义。描述性集合理论提供了对所有可能的“真实”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。最近的研究表明，这两个历史上完全不同的逻辑分支领域之间存在着深刻的联系，而本项目旨在进一步发展这种联系。在这个项目中，PI将范畴逻辑和描述性集合理论作为无限命题和一阶逻辑的对偶“句法”和“语义”表示进行研究。在命题设置中，一长串的工作指向场所理论（“无点拓扑”）作为波兰和标准Borel空间中经典描述性集合理论的不可数概括，本项目旨在通过开发更先进的经典技术的局部对立物来扩展这种类比，例如投影集，有效描述性集合理论和强迫。在一阶设置中，PI最近的工作表明，拓扑的Joyal-Tierney表示定理产生了无限一阶逻辑理论与其可数模型的标准Borel群之间的对偶性。该项目旨在将这种对偶性扩展到度量结构的连续逻辑，这将涉及首先开发拓扑理论的度量模拟的基础。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.