A categorical approach to Gödel’s incompleteness via arithmetic universes

Yuto Ikeda CSCAT 2025, March 2025, Kumamoto, Japan

@misc{ikeda-cscat2025,
  author       = {Ikeda, Yuto},
  title        = {A Categorical Approach to {G{\"o}del}'s Incompleteness via Arithmetic Universes},
  howpublished = {Talk at CSCAT 2025},
  year         = {2025},
  month        = mar,
  url          = {https://ikeda.ac/talks/cscat2025/}
}

Abstract

In the 1970s, Joyal provided a categorical interpretation of Gödel’s incompleteness theorems in his lecture, introducing certain categories called “arithmetic universes”. Although this approach would offer a structural perspective on self-referential concepts in logic, such as incompleteness, Joyal’s insights were never formally published and have remained largely unexplored. In this talk, we present our attempt to reconstruct and refine his proof and discuss the applicability of Joyal’s idea.