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Turing degrees and many-one degrees of maximal sets

  • Manuel Lerman (a1)

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Martin [4, Theorems 1 and 2] proved that a Turing degree a is the degree of a maximal set if, and only if, a′ = 0″. Lachlan has shown that maximal sets have minimal many-one degrees [2, §1] and that every nonrecursive r.e. Turing degree contains a minimal many-one degree [2, Theorem 4]. Our aim here is to show that any r.e. Turing degree a of a maximal set contains an infinite number of maximal sets whose many-one degrees are pairwise incomparable; hence such Turing degrees contain an infinite number of distinct minimal many-one degrees. This theorem has been proved by Yates [6, Theorem 5] in the case when a = 0′.

The need for this theorem first came to our attention as a result of work done by the author [3, Theorem 2.3]. There we looked at the structure / obtained from the recursive functions of one variable under the equivalence relation f ~ g if, and only if, f(x) = g(x) a.e., that is, for all but finitely many x, where M is a maximal set, and M is its complement. We showed that /1/2 if, and only if, 1 =m2, i.e., 1. and 2 have the same many-one degree. However, it might be possible that some Turing degree of a maximal set contains exactly one many-one degree of a maximal set. Theorem 1 was proved to show that this was not the case, and hence that the theory of / is not an invariant of Turing degree.

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[1]Kleene, S. C., Introduction to metamathematics, Van Nostrand, New York, 1952.
[2]Lachlan, A. H., Many-one degrees (unpublished).
[3]Lerman, M., Recursive functions modulo co-maximal sets, Dissertation, Cornell University, Ithaca, N.Y., 1968.
[4]Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295310.
[5]Sacks, G. E., Degrees of unsolvability, Annals of mathematics studies, no. 55, Princeton, 1963.
[6]Yates, C. E. M., On the degrees of index sets. II, Transactions of the American Mathematical Society, vol. 135 (1969), pp. 249266.
[7]Yates, C. E. M., Recursively enumerable sets and retracing functions, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 8 (1962), pp. 331345.

Turing degrees and many-one degrees of maximal sets

  • Manuel Lerman (a1)

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