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Hierarchies over recursive well-orderings

Published online by Cambridge University Press:  12 March 2014

Herbert Enderton  and
David Luckham
Herbert Enderton
Affiliation:
Massachusetts Institute of Technology
David Luckham
Affiliation:
Massachusetts Institute of Technology
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Extract

In the original example of a transfinite hierarchy of degrees of unsolvability, a predicate Ha is associated with each a in the set 0 of ordinal notations. (See Kleene [K1], Spector [S1], and the references there to Davis and Mostowski.) The predicates are defined by means of induction over the partial well-ordering relation ≤0 on the set 0 of notations for the recursive ordinals. The usefulness of this hierarchy of predicates is enhanced by Spector's proof [S1] of the “uniqueness” property: viz. that the degree of Ha depends only on the ordinal |a| for which a is a notation.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 29 , Issue 4 , December 1964 , pp. 183 - 190
Copyright
Copyright © Association for Symbolic Logic 1964

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References

[E]Enderton, H., Hierarchies in recursive function thjeory, Transactions of the American Mathematical Society, vol. III (1964), pp. 457–471.CrossRefGoogle Scholar
[IM]Kleene, S. C., Introduction to Metamathematics, Van Nostrana, New York, 1952.Google Scholar
[K1]Kleene, S. C., Hierarchies of number-theoretic predicates, Bulletin of the American Mathematical Society, vol. 61 (1955) pp. 193–213.CrossRefGoogle Scholar
[K2]Kleene, S. C., Arithmetical predicates and function quantifiers, Transactions of the American Mathematical Society, vol. 79 (1955) pp. 312–340.CrossRefGoogle Scholar
[R]Rogers, H. Jr., Recursive functions over well ordered partial orderings, Proceedings of the American Mathematical Society, vol. 10 (1959) pp. 847–853.CrossRefGoogle Scholar
[S1]Spector, C., Recursive well-orderings, this Journal, vol. 20 (1955) pp. 151–163.Google Scholar
[S2]Spector, C., Strongly invariant hierarchies (abstract), Notices of the American Mathematical Society, vol. 5 (1958) p. 851.Google Scholar