Non-tightness in class theory and second-order arithmetic
Logic
2023-05-16 v2
Abstract
A theory T is tight if different deductively closed extensions of T (in the same language) cannot be bi-interpretable. Many well-studied foundational theories are tight, including PA [Visser2006], ZF, Z2, and KM [enayat2017]. In this article we extend Enayat's investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of Z2 and KM gives non-tight theories. Specifically, we show that GB and ACA0 each admit different bi-interpretable extensions, and the same holds for their extensions by adding Sigma^1_k-Comprehension, for k <= 1. These results provide evidence that tightness characterizes Z2 and KM in a minimal way.
Keywords
Cite
@article{arxiv.2212.04445,
title = {Non-tightness in class theory and second-order arithmetic},
author = {Alfredo Roque Freire and Kameryn J. Williams},
journal= {arXiv preprint arXiv:2212.04445},
year = {2023}
}