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Strong reducibilities in α- and β-recursion theory

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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.

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References

  1. J. K. Barwise, Admissible sets and structures (Springer, Berlin, 1975).

    CrossRef  MATH  Google Scholar 

  2. K. J. Devlin, Aspects of constructibility, Lecture Notes in Mathematics 354 (Springer, Berlin, 1973).

    MATH  Google Scholar 

  3. K. J. Devlin, Constructibility, in: J. K. Barwise (ed.). Handbook of Mathematical Logic (North-Holland, Amsterdam, 1977).

    Google Scholar 

  4. M. Dietzfelbinger, Strong reducibilities in α-and β-recursion theory (Diplomarbeit, München, 1982).

    Google Scholar 

  5. F. R. Drake, Set theory: An introduction to large cardinals (North-Holland, Amsterdam, 1974).

    MATH  Google Scholar 

  6. J. E. Fenstad, General recursion theory: an axiomatic approach (Springer, Berlin, 1980).

    CrossRef  MATH  Google Scholar 

  7. S. D. Friedman, Recursion on inadmissible ordinals, Ph.D. Thesis, M.I.T., Cambridge, MA., 1976.

    Google Scholar 

  8. S. D. Friedman, β-recursion theory, Trans. Am. Math. Soc. 255 (1979), 173–200.

    MathSciNet  Google Scholar 

  9. S. D. Friedman and G. E. Sacks, Inadmissible recursion theory, Bull. Am. Math. Soc., 83 (1977), 255–256.

    CrossRef  MathSciNet  MATH  Google Scholar 

  10. P. G. Hinman, Recursion-theoretic hierarchies (Springer, Berlin, 1978).

    CrossRef  MATH  Google Scholar 

  11. R. B. Jensen, The fine structure of the constructible hierarchy, Ann. Math. Logic, 4 (1972), 229–308.

    CrossRef  MathSciNet  MATH  Google Scholar 

  12. S. Kripke, Transfinite recursions on admissible ordinals II (abstract), J. Symb. Logic, 29 (1964), 161–162.

    Google Scholar 

  13. W. Maass, Inadmissibility, tame r.e. sets and the admissible collapse, Ann. Math. Logic, 13 (1978), 149–170.

    CrossRef  MathSciNet  MATH  Google Scholar 

  14. W. Maass, Recursively invariant β-recursion theory, Ann. Math. Logic, 21 (1981), 27–73.

    CrossRef  MathSciNet  MATH  Google Scholar 

  15. P. Odifreddi, Strong reducibilities, Bull. (New Series) Am. Math. Soc., 4 (1981), 37–86.

    CrossRef  MathSciNet  MATH  Google Scholar 

  16. H. Rogers, Jr., Theory of recursive functions and effective computability (McGraw-Hill, New York, 1977).

    MATH  Google Scholar 

  17. G. E. Sacks and S. G. Simpson, The α-finite injury method, Ann. Math. Logic, 4 (1972), 323–367.

    MathSciNet  MATH  Google Scholar 

  18. C. P. Schnorr, Rekursive Funktionen und ihre Komplexität (Teubner, Stuttgart, 1974).

    CrossRef  MATH  Google Scholar 

  19. R. A. Shore, Σn sets which are Δn-incomparable (uniformly), J. Symb. Logic, 39 (1974), 295–304.

    CrossRef  MathSciNet  MATH  Google Scholar 

  20. R. A. Shore, α-recursion theory, in: J. K. Barwise (ed.). Handbook of Mathematical Logic (North-Holland, Amsterdam, 1974).

    Google Scholar 

  21. R. A. Shore, Splitting an α-r.e. set, Trans. Am. Math. Soc., 204 (1975), 65–78.

    MathSciNet  Google Scholar 

  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. 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 

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© 1985 Springer-Verlag

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Dietzfelbinger, M., Maass, W. (1985). Strong reducibilities in α- and β-recursion theory. In: Ebbinghaus, HD., Müller, G.H., Sacks, G.E. (eds) Recursion Theory Week. Lecture Notes in Mathematics, vol 1141. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0076216

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  • DOI: https://doi.org/10.1007/BFb0076216

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-15673-4

  • Online ISBN: 978-3-540-39596-6

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