End-extensions of models of weak arithmetic from complexity-theoretic containments

Abstract

We prove that if the linear-time and polynomial-time hierarchies coincide, then every model of $\Pi_1(\mathbb{N}) + \neg \Omega_1$ has a proper end-extension to a model of $\Pi_1(\mathbb{N})$, and so $\Pi_1(\mathbb{N}) + \neg \Omega_1 \vdash \mathrm{B}\Sigma_1$. Under an even stronger complexity-theoretic assumption which nevertheless seems hard to disprove using present-day methods, $\Pi_1(\mathbb{N}) + \neg \mathrm{Exp} \vdash \mathrm{B}\Sigma_1$. Both assumptions can be modified to versions which make it possible to replace $\Pi_1(\mathbb{N})$ by $\mathrm{I}\Delta_0$ as the base theory. We also show that any proof that $\mathrm{I}\Delta_0 + \neg \exp$ does not prove a given finite fragment of $\mathrm{B}\Sigma_1$ has to be "non-relativizing", in the sense that it will not work in the presence of an arbitrary oracle.