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# Admissible ordinals and intrinsic consistency

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This paper deals with a fundamental problem in the theory of recursion on initial segments of the ordinal numbers which was originated by Kripke [4] and Platek [8]. Following Kreisel-Sacks [3], we define a notion of relative recursiveness for this abstract recursion theory which allows us to speak of a function ƒ being weakly recursive in a set A via a partial recursive function g; this makes it reasonable to speak of partial recursive functions as ‘reduction procedures’. We say that a reduction procedure g is intrinsically consistent if for any set A there is a partial function ƒ which is weakly recursive in A via g. It is an elementary result of ordinary recursion theory that an arbitrary reduction procedure can be replaced in a uniform manner by an intrinsically consistent reduction procedure without the loss of any function f being reducible to any set A. In §3 we show that for recursion theory on the ordinals less than any infinite cardinal the analogous result is consistent with the axioms of set theory. In §5 we show that for all other cases of abstract recursion theory the analogous result fails in a very strong way to be true.

## References

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[1]Cohen, P. J., Set theory and the continuum hypothesis, W. A. Benjamin, Inc., New York, 1966.
[2]Kleene, S. C., Introduction to metamathematics, Van Nostrana, New York, 1952.
[3]Kreisel, G. and Sacks, G. E., Metarecursive sets, this Journal, vol. 31 (1966), pp. 121.
[4]Kripke, S., Transfinite recursions on admissible ordinals, I, II, abstracts, this Journal, vol. 29 (1964), pp. 161162.
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[7]Owings, J. C., Topics in metarecursion theory, Ph.D. thesis, Cornell University, 1966.
[8]Platek, R., Foundations of recursion theory, Ph.D. thesis and supplement, Stanford University, 1966.
[9]Sacks, G. E., Post's problem, admissible ordinals, and regularity, Transactions of the American Mathematical Society, vol. 124 (1966), pp. 123.
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[12]Tharp, L. H., Set theory, mimeographed notes, M.I.T., 1966.

# Admissible ordinals and intrinsic consistency

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