Provably $Δ^0_2$ and weakly descending chains

Details Provably $Δ^0_2$ and weakly descending chains Provably $Δ^0_2$ and weakly descending chains

2010-05-12

arxiv.org/abs/1005.1989v1

Mathematics

Logic

Scientific paper

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.

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