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NON-TIGHTNESS IN CLASS THEORY AND SECOND-ORDER ARITHMETIC
Published online by Cambridge University Press: 13 June 2023
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 $\mathsf {PA}$ [39],
$\mathsf {ZF}$,
$\mathsf {Z}_2$, and
$\mathsf {KM}$ [6]. In this article we extend Enayat’s investigations to subsystems of these latter two theories. We prove that restricting the Comprehension schema of
$\mathsf {Z}_2$ and
$\mathsf {KM}$ gives non-tight theories. Specifically, we show that
$\mathsf {GB}$ and
$\mathsf {ACA}_0$ each admit different bi-interpretable extensions, and the same holds for their extensions by adding
$\Sigma ^1_k$-Comprehension, for
$k \ge 1$. These results provide evidence that tightness characterizes
$\mathsf {Z}_2$ and
$\mathsf {KM}$ in a minimal way.
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- © The Author(s), 2023. Published by Cambridge University Press on behalf of The Association for Symbolic Logic
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