Completions of Restricted Complexity I, Weak Arithmetical Theories

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.