Non-tightness in class theory and second-order arithmetic

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.