Advertisement

Strong reducibilities in α- and β-recursion theory

  • Martin Dietzfelbinger
  • Wolfgang Maass
Conference paper
  • 214 Downloads
Part of the Lecture Notes in Mathematics book series (LNM, volume 1141)

Abstract

We start an investigation of strong reducibilities in α- and β-recursion theory. In particular, we study Myhill's Theorem about recursive isomorphisms (A≤1 B & B≤1 A <=> A ≡ B), and show that it holds for a limit ordinal β if and only if σlcfβ=ω. (In particular, it fails for all admissible α>ω.) We point out a consequence for
-sets (n≥2) under V=L.

Keywords

Strong Reducibility Partial Function Isomorphism Type Acceptable Numbering Recursion Theory 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. K. Barwise, Admissible sets and structures (Springer, Berlin, 1975).CrossRefzbMATHGoogle Scholar
  2. [2]
    K. J. Devlin, Aspects of constructibility, Lecture Notes in Mathematics 354 (Springer, Berlin, 1973).zbMATHGoogle Scholar
  3. [3]
    K. J. Devlin, Constructibility, in: J. K. Barwise (ed.). Handbook of Mathematical Logic (North-Holland, Amsterdam, 1977).Google Scholar
  4. [4]
    M. Dietzfelbinger, Strong reducibilities in α-and β-recursion theory (Diplomarbeit, München, 1982).Google Scholar
  5. [5]
    F. R. Drake, Set theory: An introduction to large cardinals (North-Holland, Amsterdam, 1974).zbMATHGoogle Scholar
  6. [6]
    J. E. Fenstad, General recursion theory: an axiomatic approach (Springer, Berlin, 1980).CrossRefzbMATHGoogle Scholar
  7. [7]
    S. D. Friedman, Recursion on inadmissible ordinals, Ph.D. Thesis, M.I.T., Cambridge, MA., 1976.Google Scholar
  8. [8]
    S. D. Friedman, β-recursion theory, Trans. Am. Math. Soc. 255 (1979), 173–200.MathSciNetGoogle Scholar
  9. [9]
    S. D. Friedman and G. E. Sacks, Inadmissible recursion theory, Bull. Am. Math. Soc., 83 (1977), 255–256.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    P. G. Hinman, Recursion-theoretic hierarchies (Springer, Berlin, 1978).CrossRefzbMATHGoogle Scholar
  11. [11]
    R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic, 4 (1972), 229–308.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    S. Kripke, Transfinite recursions on admissible ordinals II (abstract), J. Symb. Logic, 29 (1964), 161–162.Google Scholar
  13. [13]
    W. Maass, Inadmissibility, tame r.e. sets and the admissible collapse, Ann. Math. Logic, 13 (1978), 149–170.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    W. Maass, Recursively invariant β-recursion theory, Ann. Math. Logic, 21 (1981), 27–73.MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    P. Odifreddi, Strong reducibilities, Bull. (New Series) Am. Math. Soc., 4 (1981), 37–86.MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    H. Rogers, Jr., Theory of recursive functions and effective computability (McGraw-Hill, New York, 1977).zbMATHGoogle Scholar
  17. [17]
    G. E. Sacks and S. G. Simpson, The α-finite injury method, Ann. Math. Logic, 4 (1972), 323–367.MathSciNetzbMATHGoogle Scholar
  18. [18]
    C. P. Schnorr, Rekursive Funktionen und ihre Komplexität (Teubner, Stuttgart, 1974).CrossRefzbMATHGoogle Scholar
  19. [19]
    R. A. Shore, Σn sets which are Δn-incomparable (uniformly), J. Symb. Logic, 39 (1974), 295–304.MathSciNetCrossRefzbMATHGoogle Scholar
  20. [20]
    R. A. Shore, α-recursion theory, in: J. K. Barwise (ed.). Handbook of Mathematical Logic (North-Holland, Amsterdam, 1974).Google Scholar
  21. [21]
    R. A. Shore, Splitting an α-r.e. set, Trans. Am. Math. Soc., 204 (1975), 65–78.MathSciNetGoogle Scholar
  22. [22]
    S. G. Simpson, Degree theory on admissible ordinals, in: J. E. Fenstad, P. G. Hinman (eds.), Generalized recursion theory (North-Holland, Amsterdam, 1974).Google Scholar
  23. [23]
    S. G. Simpson, Short course on admissible rcursion theory, in: J. E. Fenstad, G. E. Sacks (eds.), Generalized recursion theory II (North-Holland, Amsterdam, 1978).Google Scholar

Copyright information

© Springer-Verlag 1985

Authors and Affiliations

  • Martin Dietzfelbinger
    • 1
  • Wolfgang Maass
    • 1
  1. 1.Department of Mathematics, Statistics and Computer ScienceUniversity of Illinois at ChicagoChicago

Personalised recommendations