A Note On the Turing Degrees of Divergence Bounded Computable Reals

Authors:
Xizhong Zheng;Robert Rettinger
Affiliations:
Theoretische Informatik, BTU Cottbus, Germany;Theoretische Informatik II, FernUniversität Hagen, Germany
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2005

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Abstract

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 [R. Downey, G. Wu, and X. Zheng. Degrees of d.c.e. reals. Mathematical Logic Quartely, 2004. (to appear)] and show that there exists a @D"2^0-Turing degree which contains no divergence bounded computable real numbers. This extends the result of [R. Downey, G. Wu, and X. Zheng. Degrees of d.c.e. reals. Mathematical Logic Quartely, 2004. (to appear)] that not every @D"2^0-Turing degree contains a d-c.e. real.