Undecidability of local structures of $\mathrm{s}$-degrees and $\mathrm{Q}$-degrees

Abstract

We show that the first order theory of the $\Sigma^0_2\;$ $\mathrm{s}$-degrees is undecidable. Via isomorphism of the $\mathrm{s}$-degrees with the $\mathrm{Q}$-degrees, this also shows that the first order theory of the $\Pi^0_2\;$ $\mathrm{Q}$-degrees is undecidable. Together with a result of Nies, the proof of the undecidability of the $\Sigma^0_2\;$ $\mathrm{s}$-degrees yields a new proof of the known fact (due to Downey, LaForte and Nies) that the first order theory of the c.e. $\mathrm{Q}$-degrees is undecidable.