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Index sets of finite classes of recursively enumerable sets

Published online by Cambridge University Press:  12 March 2014

Louise Hay
Louise Hay*
Affiliation:
University of Illinois at Chicago Circle Mount Holyoke College
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Extract

Let q0, q1,… be a standard enumeration of all partial recursive functions of one variable. For each i, let wi = range qi and for any recursively enumerable (r.e.) set α, let θα = {n | wn = α}. If A is a class of r.e. sets, let θA = the index set of A = {n | wn ∈ A}. It is the purpose of this paper to classify the possible recursive isomorphism types of index sets of finite classes of r.e. sets. The main theorem will also provide an answer to the question left open in [2] concerning the possible double isomorphism types of pairs (θα, θβ) where α ⊂ β.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 1 , 29 May 1969 , pp. 39 - 44
Copyright
Copyright © Association for Symbolic Logic 1969

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References

[1]Dekker, J. C. E. and Myhill, J., Some theorems on classes of recursively enumerable sets, Transactions of the American Mathematical Society, vol. 89 (1958), pp. 25–59.CrossRefGoogle Scholar
[2]Hay, L., Isomorphism types of index sets of partial recursive functions, Proceedings of the American Mathematical Society, vol. 17 (1966), pp. 106–110.CrossRefGoogle Scholar
[3]Myhill, J., Creative sets, Zeitschrift für Mathematische Logik und Grundlagen der Mathematik, vol. 1 (1955), pp. 97–108.CrossRefGoogle Scholar
[4]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[5]Smullyan, R., Theory of formal systems, Annals of mathematics studies, No. 47, Princeton University Press, Princeton, N.J., 1961.CrossRefGoogle Scholar