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Journal of the London Mathematical Society 2002 66(3):513-528; doi:10.1112/S0024610702003691
© 2002 by London Mathematical Society
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© The London Mathematical Society

Turing Definability in the Ershov Hierarchy

S. Barry Cooper and Angsheng Li

Department of Pure Mathematics, School of Mathematics, University of Leeds Leeds LS2 9JT, s.b.cooper{at}leeds.ac.uk
Department of Pure Mathematics, School of Mathematics, University of Leeds Leeds LS2 9JT, angsheng{at}amsta.leeds.ac.uk
Permanent address: Institute of Software, Chinese Academy of Sciences PO Box 8718, Beijing 100080, China, liang{at}ox.ios.ac.cn

Received 22 May 2001. Revision received 13 February 2002.

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 a and any computably enumerable degree b, if b < a, then there are DCE degrees x0, x1 such that b < x0, x1 < a and a = x0 {vee} x1. The construction is unusual in that it is incompatible with upper cone avoidance.


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