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Degrees of classes of RE sets1

Published online by Cambridge University Press:  12 March 2014

J. R. Shoenfield
J. R. Shoenfield*
Affiliation:
Duke University, Durham North Carolina 27706
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Extract

In [3], Martin computed the degrees of certain classes of RE sets. To state the results succinctly, some notation is useful.

If A is a set (of natural numbers), dg(A) is the (Turing) degree of A. If A is a class of sets, dg ( A ) = {dg(A): A ∈ A ). Let M be the class of maximal sets, HHS the class of hyperhypersimple sets, and DS the class of dense simple sets. Martin showed that dg ( M ), dg ( HHS ), and dg ( DS ) are all equal to the set H of RE degrees a such that a ′ = 0″.

Let M * be the class of coinfinite RE sets having no superset in M ; and define HHS * and DS * similarly. Martin showed that dg ( DS *) = H. In [2], Lachlan showed (among other things) that dg ( M *)⊆K, where K is the set of RE degrees a such that a ″ > 0″. We will show that K ⊆ dg ( HHS *). Since maximal sets are hyperhypersimple, this gives dg ( M *) = dg ( HHS *) = K.

These results suggest a problem. In each case in which dg(A) has been calculated, the set of nonzero degrees in dg(A) is either H or K or the empty set or the set of all nonzero RE degrees. Is this always the case for natural classes A? Natural here might mean that A is invariant under all automorphisms of the lattice of RE sets; or that A is definable in the first-order theory of that lattice; or anything else which seems reasonable.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 41 , Issue 3 , September 1976 , pp. 695 - 696
Copyright
Copyright © Association for Symbolic Logic 1976

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Footnotes

1

The author would like to thank Carl Jockusch and Tony Martin for comments on a preliminary version of this paper.

References

REFERENCES

[1] Dekker, J. C. E., A theorem on hypersimple sets, Proceedings of the American Mathematical Society, vol. 5 (1954), pp. 791–796.CrossRefGoogle Scholar
[2] Lachlan, A. H., Degrees of recursively enumerable sets which have no maximal supersets, this Journal, vol. 33 (1968), pp. 431–443.Google Scholar
[3] Martin, D. A., Classes of recursively enumerable sets and degrees of unsolvability, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 12 (1966), pp. 295–310.CrossRefGoogle Scholar
[4] Robinson, R. M., A dichotomy of the recursively enumerable sets, Zeitschrift für mathematische Logik und Grundlagen der Mathematik, vol. 14 (1968), pp. 339–356.CrossRefGoogle Scholar