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A minimal pair of recursively enumerable degrees

Published online by Cambridge University Press:  12 March 2014

C. E. M. Yates
C. E. M. Yates*
Affiliation:
Cornell University, Institute for Advanced Study
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Extract

Our principal result is that there exist two incomparable recursively enumerable degrees whose greatest lower bound in the upper semilattice of degrees is 0. This was conjectured by Sacks [5]. As a secondary result, we prove that on the other hand there exists a recursively enumerable degree a < 0(1) such that for no non-zero recursively enumerable degree b is 0 the greatest lower bound of a and b.

The proof of the main theorem involves a method that we have developed elsewhere [8] to deal with situations in which a partial recursive functional may interfere infinitely often with an opposed requirement of lower priority.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 2 , June 1966 , pp. 159 - 168
Copyright
Copyright © Association for Symbolic Logic 1966

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References

REFERENCES

[1]Friedberg, R. M., Two recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of National Academy of Science, U.S.A., vol. 43 (1957), pp. 236–238.CrossRefGoogle ScholarPubMed
[2]Kleene, S. C., Introduction to metamathematics, Van Nostrana, New York, 1952.Google Scholar
[3]Mučnik, A. A., Negative answer to the problem of reducibility of the theory of algorithms (Russian), Doklady Akademii Nauk SSSR, vol. 108 (1956), pp. 194–197.Google Scholar
[4]Post, E. L., Recursively enumerable sets of positive integers and their decision-problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284–316.CrossRefGoogle Scholar
[5]Sacks, G. E., Degrees of unsolvability, Annals of Mathematics Study No. 55, Princeton, 1963.Google Scholar
[6]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 300–312.CrossRefGoogle Scholar
[7]Shoenfield, J. R., Applications of model theory of degrees of unsolvability, International Symposium on the Theory of Models, Berkeley, 1963.Google Scholar
[8]Yates, C. E. M., On the degree of index sets, to appear.Google Scholar