We study computably enumerable reals (i.e. their left cut is computably enumerable) in terms of their spectra of representations and presentations.

Abstract We study the relationship between a computably enumerable real and its presentations: ways of approximating the real by enumerating a prefix-free set of binary strings.

The Turing degree of a real number x is defined as the Turing degree of its binary expansion. This definition is quite natural and robust. In this paper we discuss some basic degree properties of semi-computable and weakly computable real numbers introduced by Weihrauch and Zheng [19]. Among others we show that, there are two real numbers of c.e. binary expansions such that their difference does not have an ω.c.e. Turing degree. 1

A real α is called a c. e. real if it is the halting probability of a prefix free Turing machine. Equivalently, α 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 the d. c. e. reals, which are of the form α − β,whereα 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 ∆ 0 2 degrees that are not even n-c. e. for any n and yet contain d. c. e. reals, where a degree is n-c. e. if it contains an n-c. e. set. 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 ∆ 0 2 degrees, containing no d. c. e. real. 1

The Turing degree of a real number is defined as the Turing degree of its binary expansion. In this note we apply the double witnesses technique recently developed by Downey, Wu and Zheng [2] and show that there exists a ∆0 2-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆0 2-Turing degree contains a d-c.e. real.

In a later section, we will look at a result of Coles, Downey and Slaman [16] of pure computability theory. The result is that, for any set X, the set