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A note on the Turing degree of divergence bounded computable real numbers
 CCA 2004, August 1620, Lutherstadt
, 2004
"... 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 2Turing degree which contains no divergence bounded computable real numbers ..."
Abstract

Cited by 3 (1 self)
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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 2Turing degree which contains no divergence bounded computable real numbers. This extends the result of [2] that not every ∆0 2Turing degree contains a dc.e. real.
Degrees of d.c.e. reals
 Mathematical Logic Quartely
, 2004
"... 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 t ..."
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Cited by 2 (2 self)
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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 nc. e. for any n and yet contain d. c. e. reals, where a degree is nc. e. if it contains an nc. e. set. In this paper we will prove that every ωc. e. degree contains a d. c. e. real, but there are ω +1c. e. degrees and, hence ∆ 0 2 degrees, containing no d. c. e. real. 1