We show that the first order theory of the å₂⁰ s-degrees is undecidable. Via isomorphism of the s-degrees with the Q-degrees, this also shows that the first order theory of the Õ₂⁰ Q-degrees is undecidable. Together with a result of Nies, the proof of the undecidability of the å₂⁰ 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. Q-degrees is undecidable.

Tbilisi Mathematical Journal, Vol. 1(2008), pp. 15-32