Completions of Restricted Complexity I, Weak Arithmetical Theories

TL;DR

This paper investigates the nature of complete theories of arithmetic with restricted complexity, constructing models where such theories are of limited logical complexity, revealing insights into the structure of arithmetical theories.

Contribution

It constructs models of arithmetic with complete theories of restricted complexity, including a model of pen + Coll with this property, advancing understanding of arithmetical theory completions.

Findings

No consistent extension of delta_0 + Exp has a restricted complexity completion.

Models of arithmetic with complete theories of restricted complexity exist.

A model of pen + Coll has a complete theory of restricted complexity.

Abstract

Given a first-order theory $T$ formulated in the usual language of first-order arithmetic, we say that $T$ is of *restricted complexity* if there is some natural number $n$ and some set $\mathcal A$ of $\Sigma_n$-sentences such that $T$ can be axiomatized by $\mathcal A$. Motivated by the fact that no consistent arithmetical theory extending $\mathrm{I}\Delta _{0}+\mathsf{Exp}$ has a consistent completion that is of restricted complexity, we construct models of arithmetic whose complete theories are of restricted complexity. Our strongest result shows that there is a model of $\mathsf{IOpen + Coll}$ whose complete theory is of restricted complexity, where $\mathsf{Coll}$ is the full collection scheme.