In a recent paper, Enayat and Łełyk [2024] show that second order arithmetic and countable set theory are not definitionally equivalent. It is well known that these theories are biinterpretable. Thus, we have a pair of natural theories that illustrate a meaningful difference between definitional equivalence and bi-interpretability. This is particularly interesting given that Visser and Friedman [2014] have shown that a wide class of natural foundational theories in mathematics are such that if they are bi-interpretable, then they are also definitionally equivalent. The proof offered by Enayat and Łełyk makes use of an inaccessible cardinal. In this short note, we show that the failure of bi-interpretability can be established in Peano Arithmetic merely supposing that one of our target theories are consistent.
@misc{chenTeasingApartDefinitional2025,title={Teasing Apart Definitional Equivalence},author={Chen, Jason and Meadows, Toby},year={2025},publisher={arXiv},doi={10.48550/ARXIV.2508.03956},urldate={2025-08-07},copyright={Creative Commons Attribution 4.0 International},keywords={FOS: Mathematics,Logic (math.LO)}}
2024
PhD Dissertation
Confluence and Classification: Towards a Philosophy of Descriptive-Set-Theoretic Practice
@phdthesis{chen2024confluence,title={Confluence and Classification: Towards a Philosophy of Descriptive-Set-Theoretic Practice},author={Chen, Jason Zesheng},year={2024},school={University of California, Irvine},}