Descriptive Set Theory and 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.