Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-25T18:34:48.214Z Has data issue: false hasContentIssue false
  • English
  • Français

Transfinite extensions of Friedberg's completeness criterion1

Published online by Cambridge University Press:  12 March 2014

John M. Macintyre
Get access Rights & Permissions [Opens in a new window]

Extract

In [3] Friedberg showed that every Turing degree ≥ 0′ is the jump of some degree. Using the relativized version of this theorem it can be shown by finite induction that if a ≥ 0(n)0(n) then there is a b such that b(n) = a (our notation is defined in §1). It is natural to ask whether these results can be extended into the transfinite. Is it true for example, that whenever a ≥ 0(ω) there is a b such that = b(ω)= a? In §2 we use forcing to prove this result. (The usefulness of forcing in answering questions involving the jump operation on degrees of unsolvability has been previously demonstrated in Selman [5] where, for example, forcing was used to construct degrees a and b such that a(ω) = b(ω) = a ∪ b = 0(ω).) In §3 we generalize the methods of §2 to show that if α is a recursive ordinal and a ≥ 0(α) then there is a bsuch that b(α) = a, i.e. the Friedberg result can be extended to all recursive ordinal levels.

Thomason [6] used a forcing argument to show: If (the Kleene set of notations for the recursive ordinals) then there is a B such that (the set of notations for ordinals recursive in B). In §4 we show this result holds when hyperarithmetic reducibility is replaced by Turing reducibility: If then there is a B such that .

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 42 , Issue 1 , March 1977 , pp. 1 - 10
Copyright
Copyright © Association for Symbolic Logic 1977

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

1

This work was partially supported by NSF Grant GP–29079.

References

BIBLIOGRAPHY

[1]Feferman, S., Some applications of the notion of forcing and generic sets, Fundamente Mathematicae, vol. 56 (1956), pp. 325–345.CrossRefGoogle Scholar
[2]Feferman, S. and Spector, C., Incompleteness along paths in progressions of theories, this Journal, vol. 27 (1962), pp. 383–90.Google Scholar
[3]Friedberg, R. M., A criterion for completeness of degrees of unsolvability, this Journal, vol. 22 (1957), pp. 159–60.Google Scholar
[4]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[5]Selman, A. L., Applications of forcing to the degree theory of the arithmetic hierarchy, The Journal of the London Mathematical Society, vol. 25 (1973), pp. 586–602.Google Scholar
[6]Thomason, S. K., The forcing method and the upper semilattice of degrees, Transactions of the American Mathematical Society, vol. 129 (1967), pp. 38–57.CrossRefGoogle Scholar