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：范畴逻辑和描述集合论。广义地说，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”根据“数字”的环境(整数、小数等)的不同而改变含义。一个是指。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，对数学语句的解释同时捕获了它们在所有可能的“真实”宇宙中的含义。描述性集合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。最近的工作指出了这两个历史上不同的逻辑子领域之间的深刻联系，本项目旨在进一步发展这种联系。在这个项目中，PI研究了范畴逻辑和描述集合论作为无限命题逻辑和一阶逻辑的双重“句法”和“语义”表示。在命题背景下，一长串工作指向场所理论(“无点拓扑”)，它是经典描述集合论在波兰空间和标准Borel空间中的不可数推广，本项目旨在通过发展更先进的经典技术的局部对应来扩展这种类比，如射影集合、有效描述集合论和强迫。在一阶情形下，PI最近的工作表明，拓扑点的JoYal-Tierney表示定理在无穷一阶逻辑中的理论和它们的可数模型的标准Borel群胚之间产生了对偶。该项目旨在将这种二元性扩展到公制结构的连续逻辑，这将涉及首先为公制模拟TOOS理论开发基础。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.