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Minimal α-recursion theoretic degrees

Published online by Cambridge University Press:  12 March 2014

John M. MacIntyre
John M. MacIntyre*
Affiliation:
Massachusetts Institute of Technology, Cambridge, Massachusetts 02139
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Extract

This paper investigates the problem of extending the recursion theoretic construction of a minimal degree to the Kripke [2]-Platek [5] recursion theory on the ordinals less than an admissible ordinal α, a theory derived from the Takeuti [11] notion of a recursive function on the ordinal numbers. As noted in Sacks [7] when one generalizes the recursion theoretic definition of relative recursiveness to α-recursion theory for α > ω the two usual definitions give rise to two different notions of reducibility. We will show that whenever α is either a countable admissible or a regular cardinal of the constructible universe there is a subset of α whose degree is minimal for both notions of reducibility. The result is an excellent example of a theorem of ordinary recursion theory obtainable via two different constructions, one of which generalizes, the other of which does not. The construction which cannot be lifted to α-recursion theory is that of Spector [10]. We sketch the reasons for this in §3.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 38 , Issue 1 , March 1973 , pp. 18 - 28
Copyright
Copyright © Association for Symbolic Logic 1973

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References

BIBLIOGRAPHY

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