This is a talk as the 2023 North American Annual Meeting for the Association for Symbolic Logic.
A theory $T$ is tight if different deductively closed extensions of $T$ (in the same language) cannot be bi-interpretable. Enayat showed that ZF and KM (the Kelley–Morse theory of classes with full comprehension) are tight. A question was then, is the full strength of these theories necessary for this result? In this talk I will provide evidence toward a positive answer for the case of KM. Namely, GB (the Gödel–Bernays theory of classes with only predicative comprehension) and the restriction of KM to $\Sigma^1_k$-Comprehension, for any $k$, all fail to be tight. The main idea of the construction is that minimal models of these theories are bi-interpretable with extensions by a carefully chosen Cohen generic class of ordinals.
This talk is about joint work with Alfredo Roque Freire.