Teasing apart definitional equivalence

TL;DR

This paper demonstrates that the failure of bi-interpretability between certain foundational theories can be proven within Peano Arithmetic assuming only their consistency, highlighting a nuanced distinction in interpretability notions.

Contribution

It shows that the non-bi-interpretability of second order arithmetic and countable set theory can be established without large cardinal assumptions, using only consistency assumptions in Peano Arithmetic.

Findings

Bi-interpretability fails between second order arithmetic and countable set theory.

Failure of bi-interpretability can be proven in Peano Arithmetic assuming consistency.

Distinction between bi-interpretability and definitional equivalence is clarified.

Abstract

In a recent paper, Enayat and Le lyk [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 llustrate 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 Le lyk 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.