Non-tightness in class theory and second-order arithmetic

Alfredo Roque Freire; Kameryn J. Williams

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arXiv · 2212.04445

Non-tightness in class theory and second-order arithmetic

Alfredo Roque Freire·Kameryn J. Williams

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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.

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BibTeX RIS

Alfredo Roque Freire, Kameryn J. Williams. 2022-12-08. Non-tightness in class theory and second-order arithmetic. https://arxiv.org/abs/2212.04445

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