# Tightness in second-order arithmetic

This is a talk in the CUNY Models of Peano Arithmetic seminar on 2022 October 18

[slides]

Say that a theory $T$ is tight if any two distinct extensions of $T$ cannot be bi-interpretable. Vaguely speaking, tightness expresses a sort of maximality to the expressiveness of $T$. Visser showed that $\mathsf{PA}$ is tight and building on this work, Enayat showed that $\mathsf{Z}_2$, second-order arithmetic with full second-order comprehension, is also tight. In this talk I will address the question of whether full logical strength of these theories of arithmetic are necessary to have tightness, focusing on subsystems of $\mathsf{Z}_2$. The answer to this question is positive. If you restrict the comprehension axiom of $\mathsf{Z}_2$ to only arithmetical formulae, or if you restrict it to $\Sigma^1_k$ formulae, the resulting theory is not tight. As a specific instance, we show that if $M$ is either the minimum omega-model of $\mathsf{ACA}_0$ or the minimum beta-model of $\Pi^1_k$-$\mathsf{CA}_0$ for some $k \ge 1$, then $M$ is bi-interpretable with a carefully chosen extension $M[c] $ by Cohen-forcing.

This talk is about joint work with Alfredo Roque Freire.