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 ${\mathrm{\Sigma }}_n$-sentences such that $T$ can be axiomatized by $\mathcal{A}$. Motivated by the fact that no consistent arithmetical theory extending $\mathrm{I}{\mathrm{\Delta }}_0+\mathsf{E}\mathsf{x}\mathsf{p}$ 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 $\mathsf{I}\mathsf{O}\mathsf{p}\mathsf{e}\mathsf{n}+\mathsf{C}\mathsf{o}\mathsf{l}\mathsf{l}$ whose complete theory is of restricted complexity, where $\mathsf{C}\mathsf{o}\mathsf{l}\mathsf{l}$ is the full collection scheme.