Completions of Restricted Complexity I, Weak Arithmetical Theories
Ali Enayat, Mateusz Łełyk, Albert Visser
TL;DR
The paper introduces the notion of restricted complexity for arithmetical theories and proves that certain weak theories admit completions of restricted complexity, notably ${ m PA}^{-}$ and ${ m IOpen}+{ m Coll}$. It develops a framework of interpretability and partial isomorphisms to witness restricted completions, using a sequence of algebraic model constructions: ${ m Z}[ m X]^{ ext{≥0}}$, Shepherdson’s ${ m S}$, the Dorroh-augmented rings ${ m A}$, and the iterated Shepherdson models ${ m B}$. Each construction yields a completion axiomatizable by a single sentence plus a (potentially infinite) Σ₁-sentences set, thereby separating restricted-complexity completions from other complexity classes (MRDP, truth, nrec). The results illuminate how bounded-quantifier and real-closure techniques can be leveraged to control completeness while maintaining strong algebraic properties, and they set the stage for analogous results in stronger theories and higher-order contexts. The work advances our understanding of incompleteness phenomena by showing that restricted-complexity completions can exist even for theories with full Induction-like strength when coupled with certain collection principles.
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 $Σ_n$-sentences such that $T$ can be axiomatized by $\mathcal A$. Motivated by the fact that no consistent arithmetical theory extending $\mathrm{I}Δ_{0}+\mathsf{Exp}$ 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.