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Uniform enumeration operations

Published online by Cambridge University Press:  12 March 2014

A. H. Lachlan*
Affiliation:
Simon Fraser University, Burnaby 2, British Columbia, Canada

Abstract

Sacks [2] has asked whether there exists a uniform solution to Post's problem, i.e. an enumeration operation W such that d < W(d) < d′ for every degree d. It is shown here that if such an operation W exists it cannot itself in a particular technical sense be uniform. In fact, the jump operation is characterized amongst such uniform enumeration operations by the condition: d < W(d) for all d. In addition, it is proved that the only other uniform enumeration operations such that dW(d) for all d are those which equal the identity operation above some fixed degree.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 1975

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References

REFERENCES

[1]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[2]Sacks, G. E., On a theorem of Lachlan and Martin, Proceedings of the American Mathematical Society, vol. 18 (1967), pp. 140141.CrossRefGoogle Scholar