Skip to main content
SpringerLink
Log in

Limit lemmas and jump inversion in the enumeration degrees

  • Published:
Archive for Mathematical Logic Aims and scope Submit manuscript

Abstract.

We show that there is a limit lemma for enumeration reducibility to 0 e ', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X< e W such that W≤ e X' there is a set A such that X≤ e A< e W and A'=W'. (All jumps are enumeration degree jumps.) The degrees of sets with good approximations include the Σ0 2 degrees and the n-CEA degrees.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Calhoun, W.C., Slaman, T.A.: The Π0 2 enumeration degrees are not dense. Journal of Symbolic Logic 61, 1364–1379 (1996)

    MathSciNet  MATH  Google Scholar 

  2. Cooper, S.B.: Partial degrees and the density problem, part ii: the enumeration degrees of σ2 sets are dense. Journal of Symbolic Logic 49, 503–513 (1984)

    MathSciNet  MATH  Google Scholar 

  3. Cooper, S.B.: Enumeration reducibility, nondeterministic computations and relative computability of partial functions. In Ambos-Spies K., Müller G.H., and Sacks G.E. (editors), Recursion Theory Week, Proceedings Oberwolfach, Lecture Notes in Mathemathics, 1432, 57–110. Springer-Verlag, Berlin Heidelberg New York, 1989

    Google Scholar 

  4. Cooper, S.B., Copestake, C.S.: Properly §igma 2 enumeration degrees. Zeitschrift für mathematische Logik und Grundlagen der Mathematik, 34, 491–522 (1988)

    Google Scholar 

  5. Lachlan, A.H., Shore, R.A.: The n-rea enumeration degrees are dense. Archive for Mathematical Logic 31, 277–285 (1992)

    MathSciNet  MATH  Google Scholar 

  6. McEvoy, K.: Jumps of quasi-minimal enumeration degrees. Journal of Symbolic Logic 50, 839–848 (1985)

    MathSciNet  MATH  Google Scholar 

  7. McEvoy, K., Cooper, S.B.: On minimal pairs of enumeration degrees. Journal of Symbolic Logic 50, 983–1001 (1985)

    MathSciNet  MATH  Google Scholar 

  8. Rogers, H.: Theory of recursive functions and effective computability. New York: McGraw-Hill, 1967

  9. Sacks, G.E.: Recursive enumerability and the jump operator. Transactions of the American Mathematical Society 108, 223–239 (1963)

    Google Scholar 

  10. Sorbi, A.: The enumeration degrees of the §igma 0 2 sets. In Lecture Notes in Pure and Applied Math., 187, 303–330. Dekker, New York, 1997

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. School of Mathematical and Computing Sciences, Victoria University of Wellington, PO Box 600, Wellington, New Zealand

    Evan J. Griffiths

Authors
  1. Evan J. Griffiths
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Evan J. Griffiths.

Additional information

The results in this paper form part of the author's doctoral dissertation written under the supervision of Prof. Steffen Lempp at the University of Wisconsin Madison. The author is grateful to an anonymous referee for helpful comments and suggestions.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Griffiths, E. Limit lemmas and jump inversion in the enumeration degrees. Arch. Math. Logic 42, 553–562 (2003). https://doi.org/10.1007/s00153-002-0161-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00153-002-0161-z

Keywords

Search

Navigation