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A note on degrees of subsets1

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

In [2] we constructed an infinite set of natural numbers containing nosubset of higher (Turing) degree. Since it is well knownthat there are nonrecursive sets (e.g. sets of minimal degree) containing nononrecursive subset of lower degree, it is natural tosuppose that these arguments may be combined, but this is false. We provethat every infinite set must contain a nonrecursive subset of either higheror lower degree.

Information

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 34 , Issue 2 , 25 July 1969 , pp. 256
Copyright
Copyright © Association for Symbolic Logic 1969

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Footnotes

1

This work was supported by National Science Foundation Grant GP8866.

References

[1] Rogers, H. Jr., Theory of recursive functions and effective computability, McGraw-Hill, New York, 1967.Google Scholar
[2] Soare, R. I., Sets with no subset of higher degree, this Journal , vol. 34, (1969), pp. 53–56.Google Scholar