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The impossibility of finding relative complements for recursively enumerable degrees

Published online by Cambridge University Press:  12 March 2014

A. H. Lachlan
A. H. Lachlan*
Affiliation:
University of Newcastle Upon Tyne, Simon Frazer University
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Extract

J. R. Shoenfield conjectured in a talk at the Berkeley Model Theory Symposium (1963) that, if b and d are non-zero recursively enumerable (r.e.) degrees such that b < d then there exists an r.e. degree c such that c < d and b U c = d. G. E. Sacks echoed this conjecture at the end of [3]. In this paper the conjecture is disproved. We construct r.e. degrees b, d such that 0 < b < d and such that for no r.e. degree c is it true that c < d and b U c = d. We are grateful to G. E. Sacks for suggesting this problem.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 31 , Issue 3 , 02 September 1966 , pp. 434 - 454
Copyright
Copyright © Association for Symbolic Logic 1996

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References

[1]Kleene, S. C., Introduction to Metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952, x + 550 p.Google Scholar
[2]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
[3]Sacks, G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), 300–312.CrossRefGoogle Scholar