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TURING DEFINABILITY IN THE ERSHOV HIERARCHY

Published online by Cambridge University Press:  24 March 2003

S. BARRY COOPER
Affiliation:
Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT s.b.cooper@leeds.ac.uk, angsheng@amsta.leeds.ac.uk
ANGSHENG LI
Affiliation:
Department of Pure Mathematics, School of Mathematics, University of Leeds, Leeds LS2 9JT s.b.cooper@leeds.ac.uk, angsheng@amsta.leeds.ac.uk Permanent address: Institute of Software, Chinese Academy of Sciences, PO Box 8718, Beijing 100080, Chinaliang@ox.ios.ac.cn
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Abstract

The first nontrivial DCE (2-computably enumerable) Turing approximation to the class of computably enumerable degrees is obtained. This depends on the following extension of the splitting theorem for the DCE degrees. For any DCE degree ${\bf a}$ and any computably enumerable degree ${\bf b}$ , if ${\bf b} < {\bf a}$ , then there are DCE degrees ${\bf x_0}, {\bf x_1}$ such that ${\bf b} < {\bf x_0}, {\bf x_1} < {\bf a}$ and ${\bf a} = {\bf x_0} \lor {\bf x_1}$ . The construction is unusual in that it is incompatible with upper cone avoidance.

Type
Notes and Papers
Copyright
© The London Mathematical Society, 2002

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