Advertisement

Siberian Mathematical Journal

, Volume 51, Issue 6, pp 1135–1138 | Cite as

Computable numberings of families of low sets and Turing jumps in the Ershov hierarchy

  • M.Kh. FaizrahmanovEmail author
Article
  • 22 Downloads

Abstract

If ν and μ are some Δ 2 0 -computable numberings of families of sets of the naturals then P(x,y) ⇔ ν(x)′ ≠ μ(y) is a Σ 2 0 -predicate. Deriving corollaries from this result, we obtain a sufficient condition for existence of a Δ 2 0 -computable numbering of the subfamily of all sets in a given family with the Turing jumps belonging to a fixed level of the Ershov hierarchy, and we deduce existence of a Σ ω −1 -computable numbering of the family of all superlow sets.

Keywords

computable numbering Ershov hierarchy constructive ordinal low set superlow set 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ershov Yu. L., “A hierarchy of sets. I,” Algebra and Logic, 7, No. 3, 25–43 (1968).CrossRefGoogle Scholar
  2. 2.
    Ershov Yu. L., “A hierarchy of sets. II,” Algebra and Logic, 7, No. 4, 212–232 (1968).CrossRefGoogle Scholar
  3. 3.
    Ershov Yu. L., “On a hierarchy of sets. III,” Algebra and Logic, 9, No. 1, 20–31 (1970).zbMATHCrossRefGoogle Scholar
  4. 4.
    Arslanov M. M, The Ershov Hierarchy [in Russian], Kazan Univ., Kazan (2007).Google Scholar
  5. 5.
    Ershov Yu. L., Theory of Numberings [in Russian], Nauka, Moscow (1977).Google Scholar
  6. 6.
    Carstens H. G., “Δ20-mengen,” Arch. Math. Log. Grundlagenforsch., Bd 18, 55–65 (1976).zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Nies A., “Reals which compute little,” Lect. Notes Logic, 27, 261–275 (2002).MathSciNetGoogle Scholar
  8. 8.
    Yates C. E. M., “On the degrees of index sets. II,” Trans. Amer. Math. Soc., 135, 249–266 (1969).zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Faizrahmanov M. Kh., “Turing jumps in the Ershov hierarchy,” Algebra and Logic (to appear).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Kazan Federal UniversityKazanRussia

Personalised recommendations