The -spectrum of a linear order

Abstract

AbstractSlaman and Wehner have constructed structures which distinguish the computable Turing degree 0 from the noncomputable degrees, in the sense that the spectrum of each structure consists precisely of the noncomputable degrees. Downey has asked if this can be done for an ordinary type of structure such as a linear order. We show that there exists a linear order whose spectrum includes every noncomputable degree, but not 0. Since our argument requires the technique of permitting below a set, we include a detailed explantion of the mechanics and intuition behind this type of permitting.