Archive for Mathematical Logic

, Volume 46, Issue 7–8, pp 565–582 | Cite as

Effectively closed sets and enumerations

  • Paul Brodhead
  • Douglas Cenzer


An effectively closed set, or \({\Pi^{0}_{1}}\) class, may viewed as the set of infinite paths through a computable tree. A numbering, or enumeration, is a map from ω onto a countable collection of objects. One numbering is reducible to another if equality holds after the second is composed with a computable function. Many commonly used numberings of \({\Pi^{0}_{1}}\) classes are shown to be mutually reducible via a computable permutation. Computable injective numberings are given for the family of \({\Pi^{0}_{1}}\) classes and for the subclasses of decidable and of homogeneous \({\Pi^{0}_{1}}\) classes. However no computable numberings exist for small or thin classes. No computable numbering of trees exists that includes all computable trees without dead ends.

Mathematics Subject Classification (2000)

03D30 03D25 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Binns, S.: Small \({\Pi^{0}_{1}}\) classes. Arch. Math. Logic 45(4), 393–410 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Brodhead, P.: Enumerations of \({\Pi^{0}_{1}}\) classes: acceptability and decidable classes. In: Proceedings of CCA 2006, Gainesville, Elect. Notes in Th. Comp. Sci., Elsevier, Amsterdam (2006) (to appear, electronic)Google Scholar
  3. 3.
    Brodhead, P.: Computable aspects of closed sets. Ph.D. Dissertation, University of Florida (2008)Google Scholar
  4. 4.
    Cholak, P., Coles, R., Downey, R., Hermann, E.: Automorphisms of the lattice of \({\Pi^{0}_{1}}\) classes: perfect thin classes and anc degrees. Trans. Am. Math. Soc. 353, 4899–4924 (2001) (electronic)zbMATHCrossRefGoogle Scholar
  5. 5.
    Cenzer, D., Downey, R., Jockusch, C., Shore, R.: Countable thin \({\Pi^{0}_{1}}\) classes. Ann. Pure App. Logic 59, 79–139 (1993)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Cenzer, D.: \({\Pi^{0}_{1}}\) classes in computability theory. In: Griffor, E.R.(eds) Handbook of Computability Theory, pp. 37–85. Elsevier, Amsterdam (1999)CrossRefGoogle Scholar
  7. 7.
    Cenzer, D., Remmel, J.B.: Effectively Closed Sets, ASL Lecture Notes in Logic (to appear)Google Scholar
  8. 8.
    Cenzer, D., Remmel, J.: Index sets for \({\Pi^{0}_{1}}\) classes. Ann. Pure App. Logic 93, 3–61 (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    Cenzer, D., Remmel, J.: \({\Pi^{0}_{1}}\) classes in mathematics. In: Ershov, Y., Goncharov, S., Nerode, A., Remmel, J.(eds) Handbook of Recursive Mathematics, pp. 623–821. North-Holland, Amsterdam (1999)Google Scholar
  10. 10.
    Cenzer, D., Remmel, J.: Index sets for computable real functions. In: Proceedings of CCA 2003, Cincinnati, pp. 163–182 (2003)Google Scholar
  11. 11.
    Downey, R., Jockusch, C., Stob, M.: Array nonrecursive sets of multiple permitting arguments. In: Ambos-Spies, K., Muller, G., Sacks, G.(eds) Recursion Theory Week: Proc. Ober. 1989, pp. 141–173. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  12. 12.
    Downey, R.: Maximal theories. Ann. Pure App. Logic 33, 245–282 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ershov, Y.: Theory of numberings. In: Griffor, E.R.(eds) Handbook of Computability Theory, pp. 473–503. North-Holland, Amsterdam (1999)CrossRefGoogle Scholar
  14. 14.
    Friedberg, R.: Three theorems on recursive enumeration. J. Symb. Logic 23, 309–316 (1958)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Goncharov, S., Lempp, S., Solomon, R.: Friedburg numberings of families of n-computably enumerable sets. Algebra Logic 41, 81–86 (2002)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Hinman, P.: Recursion-Theoretic Hierarchies. Springer, Heidelberg (1978)zbMATHGoogle Scholar
  17. 17.
    Jockusch, C., Soare, R.: \({\Pi^{0}_{1}}\) classes and degrees of theories. Trans. Am. Math. Soc. 173, 35–56 (1972)CrossRefMathSciNetGoogle Scholar
  18. 18.
    Lempp, S.: Hyperarithmetical index sets in recursion theory. Trans. Am. Math. Soc. 303, 559–583 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Pour-El, M., Putnam, H.: Recursively enumerable classes and their application to recursive sequences of formal theories. Archiv Math. Logik Grund 8, 104–121 (1965)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Raichev, A.: Relative Randomness via RK-Reducibility, Ph.D. Thesis, University of Wisconsin, Madison (2006)Google Scholar
  21. 21.
    Rogers, H.: Gödel numberings of partial recursive functions. J. Symb. Logic 23, 331–341 (1958)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Rogers, H.: Theory of recursive functions and effective computability. McGraw-Hill, New York (1967)zbMATHGoogle Scholar
  23. 23.
    Simpson, S.: Mass problems and randomness. Bull. Symb. Logic 11(1), 1–27 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  24. 24.
    Soare, R.: Recursively Enumerable Sets and Degrees. Springer, Heidelberg (1987)Google Scholar
  25. 25.
    Solomon, R.: Thin classes of separating sets. Contemporary mathematics 425, pp 67–86. American Mathematical Society (2007)Google Scholar
  26. 26.
    Suzuki, Y.: Enumerations of recursive sets. J. Symb. Logic 24, 311 (1959)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of FloridaGainesvilleUSA

Personalised recommendations