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Sets with no subset of higher degree1

Published online by Cambridge University Press:  12 March 2014

Robert I. Soare
Robert I. Soare*
Affiliation:
University of Illinois at Chicago Circle
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Extract

The problem of finding an infinite set of natural numbers which contains no subsets of higher (Turing) degree was first posed by W. Miller [3] and was brought to our attention by C. G. Jockusch, Jr., who proved that such a set, if it existed, could not be hyperarithmetic.2 In this paper we construct an infinite set which is not recursive in any of its coinfinite subsets, and thus contains no subset of higher degree. Our original proof made use of the result (attributed to Ehrenfeucht) that every subset of 2ω which is open (in the standard topology) is “Ramsey”.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 1 , 29 May 1969 , pp. 53 - 56
Copyright
Copyright © Association for Symbolic Logic 1969

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Footnotes

1

This work was partly supported by National Science Foundation Grant GP8866. We are very grateful to C. G. Jockusch, Jr. for corrections, for several helpful discussions on the problem, and for supplying relevant information. We are also grateful to F. Galvin for telling us immediately of his results. An infinite set not recursive in any coinfinite subset was constructed simultaneously and independently by P. J. Cohen (unpublished).

References

[1]Gandy, R. O. and Sacks, G. E., A minimal hyperdegree, Fundamenta Mathematicae, vol. 61 (1967), pp. 215–223.CrossRefGoogle Scholar
[2]Jockuschi, C. G. Jr., Uniformly introreducible sets, this Journal vol. 33 (1968), pp. 521–536.Google Scholar
[3]Miller, W., Sets of integers and degrees of unsolvability, Master's thesis, University of Washington.Google Scholar
[4]Prikry, K. and Galvin, F., A combinatorial theorem (to appear).Google Scholar
[5]Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[6]Sacks, G. E., Degrees of unsolvability, Annals of Mathematics Studies, No. 55, Princeton Univ. Press, Princeton, N.J., 1963.Google Scholar
[7]Soare, R. I., Sets with no subset of higher degree, Notices of the American Mathematical Society, vol. 15 (1968), p. 513.Google Scholar
[8]Soare, R. I., Sets with no subsets of higher arithmetic degree, Notices of the American Mathematical Society, vol. 15 (1968), p. 644.Google Scholar