Degrees of d. c. e. reals

A real is called c.e. if it is the halting probability of a prefix free Turing machine. Equivalently, a real is c.e. if it is left computable in the sense that L(α) = {q ∈ Q : q ≤ α} is a computably enumerable set. The natural field formed by the c.e. reals turns out to be the field formed by the collection of reals of the form α− β where α and β are c.e. reals. While c.e. reals can only be found in the c.e. degrees, Zheng has proven that there are ∆2 degrees that are not even n-c.e. for any n and yet contain d.c.e. reals. In this paper we will prove that every ω-c.e. degree contains a d.c.e. real, but there are ω+1-c.e. degrees and, hence ∆2 degrees, containing no d.c.e. real. ∗Downey is partially supported by the New Zealand Marsden Fund. Wu is supported by the New Zealand FRST Post-Doctoral Fellowship. Downey and Wu are partially supported by the International Joint Project No. 00310308 of NSFC of China.