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A continuous functional with noncollapsing hierarchy

Published online by Cambridge University Press:  12 March 2014

Dag Normann
Dag Normann*
Affiliation:
University of Oslo, Blindern, Oslo-3, Norway
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Extract

In [5] S. S. Wainer introduces a hierarchy for arbitrary type-2-functionals. Given F, he defines a set of ordinal notations OF, and for each a ∈ OF, a function fa recursive in F and an ordinal ∣a∣F < For any f recursive in F there is an a ∈ OF such that f is primitive recursive in fa.

Let ρF be the least ordinal α such that for any f recursive in F there is an α ∈ OF with ∣a∣F ≤ α such that f is primitive recursive in fa. If ρF < the hierarchy breaks down. In Bergstra and Wainer [2] ρF is described as “the real ordinal of the 1-section of F”.

Using standard methods (originally due to Kleene) one may prove that if F is normal, then ρF = Feferman has proved that if F is recursive, then ρF = ω2.

Let 1-section (F) = l-sc(F) = {f; f is recursive in F} where f is a total object of type 1.

Grilliot [4] proved that F ↾ 1-sc(F) is continuous if and only if F is not normal.

Let h be an associate for a given functional F, and assume that h is recursive in the jump of an element of 1-sc(F).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 43 , Issue 3 , September 1978 , pp. 487 - 491
Copyright
Copyright © Association for Symbolic Logic 1978

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References

REFERENCES

[1]Bergstra, J., Computability and continuity in finite types, Dissertation, Utrecht, 1976.Google Scholar
[2]Bergstra, J. and Wainer, S., The “real” ordinal of the 1-section of a continuous functional, abstract, this Journal, vol. 42 (1977), p. 440.Google Scholar
[3]Gandy, R. O., Proof of Mostowski's conjecture, Bulletin de l'Académie Polonaise des Sciences, vol. 9 (1960), pp. 571–575.Google Scholar
[4]Grilliot, T., On effectively discontinuous type-2 objects, this Journal, vol. 36 (1971), pp. 245–248.Google Scholar
[5]Wainer, S. S., A hierarchy for the 1-section of any type two object, this Journal, vol. 39 (1974), pp. 88–94.Google Scholar
[6]Sacks, G. E., Degrees of unsolvability, Annals of mathematics studies, Princeton, 1963, 2nd edition, 1966.Google Scholar