Provably $Δ^0_2$ and weakly descending chains

In this note we show that a set is provably $Δ^0_2$ in the fragment $IΣ_n$ of arithmetic iff it is $IΣ_n$-provably in the class $D_α$ of $α$-r.e. sets in the Ershov hierarchy for an $α<_{ε_0} ω_{1+n}$, where $<_{ε_0}$ denotes a standard $ε_0$-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.

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