Descriptive Set Theory and Categorical Logic

描述集合论和分类逻辑

• 批准号: 2224709
• 负责人: Ruiyuan Chen
• 金额: $ 11.21万
• 依托单位: Regents of the University of Michigan - Ann Arbor
• 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: 2021-11-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2224709&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学逻辑的两个不同方面之间的相互作用：分类逻辑和描述集理论。 从广义上讲，数学逻辑是对数学语言及其含义的正式研究。 例如，“每两个数字之间，都有第三个变化，这意味着“数字”的环境宇宙（整数，分数等）是指的。 分类逻辑提供了构建“通用抽象宇宙”的工具，其中数学陈述的解释一次在所有可能的“真实”宇宙中捕获了它们的含义。 描述性集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种分类。 最近的工作指出了这两个历史上不同的逻辑子场之间的密切联系，该项目旨在进一步发展这种联系。在该项目中，PI研究的分类逻辑和描述性理论是双重“句法”和“语义”表示无限主词的表示和一阶逻辑。 在命题的环境中，一长串的工作指向了语言理论（“无点拓扑”），这是对波兰和标准borel空间中经典描述性理论的无数概括，并且该项目旨在通过开发更先进的经典技术的本地化对应物来扩展这种类比，例如投影式设置，有效的描述性的描述性设置，并强制强迫。 在一阶环境中，PI的最新工作表明，高级一阶逻辑中的理论与其可计数模型的标准borel glopoids中的理论之间具有双重性。 该项目旨在将这种双重性扩展到公制结构的持续逻辑，该二元性将首先开发托索斯理论的度量类似物的基础。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子优点和更广泛影响的评估标准来通过评估来获得支持的。

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
• 期刊: Theory and applications of categories
• 影响因子: 0.5
• 作者: Chen, Ruiyuan