The Slaman-Wehner theorem in higher recursion theory

Abstract

Slaman and Wehner have independently shown that there is a countable structure whose degree spectrum consists of the nonzero Turing degrees. We show that the analogue fails in the degrees of constructibility. While we do not settle the problem for the hyperdegrees, we show that every almost computable structure, in the sense of Kalimullin, has a copy computable from Kleene's O.