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.
Citation
Maria Libera Affatato. Thomas F. Kent. Andrea Sorbi. "Undecidability of local structures of $\mathrm{s}$-degrees and $\mathrm{Q}$-degrees." Tbilisi Math. J. 1 15 - 32, 2008. https://doi.org/10.32513/tbilisi/1528768822
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