Provably Δ⁰₂ and weakly descending chains

Original · EN

In this note we show that a set is provably Δ⁰₂ in the fragment IΣₙ of arithmetic iff it is IΣₙ-provably in the class Dα of α-r.e. sets in the Ershov hierarchy for an α<ε₀ ω₁₊ₙ, where <ε₀ denotes a standard ε₀-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.