Provably $\Delta^0_2$ and weakly descending chains

TL;DR

This paper characterizes provably $ ext{Delta}^0_2$ sets within fragments of arithmetic using the Ershov hierarchy and explores the strength of a limit existence rule with nested applications.

Contribution

It establishes a precise equivalence between provably $ ext{Delta}^0_2$ sets in $I ext{Sigma}_n$ and $I ext{Sigma}_n$-provability in the class $D_eta$ of the Ershov hierarchy for certain ordinals.

Findings

Characterizes $ ext{Delta}^0_2$ sets in $I ext{Sigma}_n$

Shows the strength increase of the limit existence rule with nesting

Provides a hierarchy-based classification of provable sets

Abstract

In this note we show that a set is provably $\Delta^0_2$ in the fragment $I\Sigma_n$ of arithmetic iff it is $I\Sigma_n$-provably in the class $D_\alpha$ of $\alpha$-r.e. sets in the Ershov hierarchy for an $\alpha <_{\epsilon_0} \omega_{1+n}$, where $<_{\epsilon_0}$ denotes a standard $\epsilon_0$-ordering. In the Appendix it is shown that a limit existence rule $(LimR)$ due to Beklemishev and Visser becomes stronger when the number of nested applications of the inference rule grows.