Descriptive Set Theory and Categorical Logic

描述集合论和分类逻辑

• 批准号: 2054508
• 负责人: Ruiyuan Chen
• 金额: $ 11.21万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-15 至 2022-04-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2054508&HistoricalAwards=false
• 关键词: Descriptive; Set; Theory; Categorical; Logic

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.

该项目将重点关注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含义的形式研究。 例如，断言“每两个数字之间，有第三个”的含义会根据所指的“数字”（整数、分数等）的周围宇宙而改变。 分类逻辑提供了构建“通用抽象宇宙”的工具，其中数学语句的解释立即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此类可能的“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，该项目旨在进一步发展这种联系。在该项目中，PI 研究分类逻辑和描述性集合论作为逻辑的双重“句法”和“语义”表示。无限命题和一阶逻辑。 在命题环境中，一长串的工作将语言环境理论（“无点拓扑”）视为波兰和标准 Borel 空间中经典描述集合论的不可数推广，并且该项目旨在通过开发本地对应物来扩展这种类比更先进的经典技术，例如射影集、有效描述集理论和强迫。 在一阶设置中，PI 最近的工作表明，拓扑的 Joyal-Tierney 表示定理在无限一阶逻辑理论与其可数模型的标准 Borel 群形之间产生了对偶性。 该项目旨在将这种二元性扩展到度量结构的连续逻辑，这将首先涉及为拓扑理论的度量模拟奠定基础。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点和知识进行评估，被认为值得支持。更广泛的影响审查标准。

项目成果

On sifted colimits in the presence of pullbacks

存在回调时的筛选余限

• 发表时间: 2021-12
• 期刊: Theory and applications of categories
• 作者: Chen; Ruiyuan
• 通讯作者: Ruiyuan