Archive for Mathematical Logic

, Volume 36, Issue 4–5, pp 255–267 | Cite as

Definability in the enumeration degrees

  • Theodore A. Slaman
  • W. Hugh Woodin

Abstract.

We prove that every countable relation on the enumeration degrees, \({\frak E}\), is uniformly definable from parameters in \({\frak E}\). Consequently, the first order theory of \({\frak E}\) is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order theory of the enumeration degrees of the \(\Sigma^0_2\) sets is not decidable.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Theodore A. Slaman
    • 1
  • W. Hugh Woodin
    • 1
  1. 1. Department of Mathematics, University of Chicago, Chicago, IL 60637, USA US

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