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

TL;DR

This paper explores the relationship between complexity-theoretic assumptions and models of weak arithmetic, showing how certain hierarchy coincidences imply end-extensions and proof-theoretic results, with implications for the nature of proofs involving $ ext{I} riangle_0$.

Contribution

It establishes new links between complexity assumptions and models of weak arithmetic, demonstrating end-extensions and proof limitations under these assumptions.

Findings

Models of $ ext{I} riangle_0 + eg ext{exp}$ have non-relativizing proof properties.

Hierarchy coincidences imply end-extensions of models of weak arithmetic.

Certain complexity assumptions lead to proof-theoretic consequences for $ ext{B} ext{Sigma}_1$.

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.