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Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy

  • S. S. Wainer (a1)

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It is well known that iteration of any number-theoretic function f, which grows at least exponentially, produces a new function f′ such that f is elementary-recursive in f′ (in the Csillag-Kalmar sense), but not conversely (since f′ majorizes every function elementary-recursive in f). This device was first used by Grzegorczyk [3] in the construction of a properly expanding hierarchy {ℰn: n = 0, 1, 2, …} which provided a classification of the primitive recursive functions. More recently it was shown in [7] how, by iterating at successor stages and diagonalizing over fundamental sequences at limit stages, the Grzegorczyk hierarchy can be extended through Cantor's second number-class. A problem which immediately arises is that of classifying all recursive functions, and an answer to this problem is to be found in the general results of Feferman [1]. These results show that although hierarchies of various types (including the above extensions of Grzegorczyk's hierarchy) can be produced, which range over initial segments of the constructive ordinals and which do provide complete classifications of the recursive functions, these cannot be regarded as classifications “from below”, since the method of assigning fundamental sequences at limit stages must be highly noneffective. We therefore adopt the more modest aim here (as in [7], [12], [14]) of characterising certain well-known (effectively generated) subclasses of the recursive functions, by means of hierarchies generated in a natural manner, “from below”.

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[1]Feferman, S., Classification of recursive functions by means of hierarchies, Transactions of the American Mathematical Society, vol. 104 (1962), pp. 101122.
[2]Feferman, S., Systems of predicative analysis. II, this Journal, vol. 33 (1968), pp. 193220.
[3]Grzegorczyk, A., Some classes of recursive functions, Rozprawy Matematyczne, vol. 4, 1953.
[4]Hardy, G. H., A theorem concerning the infinite cardinal numbers, Quarterly Journal of Mathematics, vol. 35 (1904), pp. 8794.
[5]Kleene, S. C., Extension of an effectively generated class of functions by enumeration, Colloquium Mathematicum, vol. 6 (1958), pp. 6778.
[6]Kreisel, G., On the interpretation of nonfinitist proofs. II, this Journal, vol. 17 (1952), pp. 4358.
[7]Löb, M. H. and Wainer, S. S., Hierarchies of number-theoretic functions. I, Archiv für matematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 3951; II, Archiv für matematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 97–113.
[8]Péter, R., Recursive functions, Academic Press, New York, 1967.
[9]Robbin, J. W., Ph.D. Dissertation, Princeton University, Princeton, N.J., 1965.
[10]Schwichtenberg, H., Eine Klassifikation der ∈0-rekursiven Funktionen, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 17 (1971), pp. 6174.
[11]Tait, W. W., Nested recursion, Mathematische Annalen, vol. 143 (1961), pp. 236250.
[12]Wainer, S. S., A classification of the ordinal recursive functions, Archiv für matematische Logik und Grundlagenforschung, vol. 13 (1970), pp. 136153.
[13]Wainer, S. S., Ph.D. Dissertation, University of Leeds, Leeds, 1969.
[14]Wainer, S. S., A subrecursive hierarchy over the predicative ordinals, Proceedings of the Conference in Mathematical Logic, Bedford College, London, 1970 (abstract to appear).

Ordinal recursion, and a refinement of the extended Grzegorczyk hierarchy

  • S. S. Wainer (a1)

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