Low upper bounds in the Turing degrees revisited

Abstract

We give an alternative proof of a result of Kucera and Slaman (2009, Lower upper bounds of ideals, Journal of Symbolic Logic, 74, 517–534) on low bounds of ideals in the Δ20 Turing degrees. This is a characterization of the ideals in the Δ20 degrees which have a low upper bound. It follows that there is a low upper bound for the ideal of the K-trivial degrees. Our proof is direct, in the sense that it does not use universal classes of PA degrees.