We show that the first order theory of the Σ2 s-degrees is undecidable. Via isomorphism of the s-degrees with the Q-degrees, this also shows that the first order theory of the Π2 Q-degrees is undecidable. Together with a result of Nies, the proof of the undecidability of the Σ2 s-degrees yields a new proof of the known fact (due to Downey, LaForte and Nies) that the first order theory of the c....

Computability theorists have studied many different reducibilities between sets of natural numbers including one reducibility (≤1), many-one reducibility (≤m), truth table reducibility (≤tt), weak truth table reducibility (≤wtt) and Turing reducibility (≤T ). The motivation for studying reducibilities stronger that Turing reducibility stems from internally motivated questions about varying the ...