NON-TIGHTNESS IN CLASS THEORY AND SECOND-ORDER ARITHMETIC

Abstract

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.