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The impossibility of finding relative complements for recursively enumerable degrees

  • A. H. Lachlan (a1)

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.

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[1]Kleene S. C., Introduction to Metamathematics, Amsterdam (North Holland), Groningen (Noordhoff), New York and Toronto (Van Nostrand), 1952, x + 550 p.
[2]Post E. L., Recursively enumerable sets of positive integers and their decision problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284316.
[3]Sacks G. E., The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), 300312.
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The Journal of Symbolic Logic
  • ISSN: 0022-4812
  • EISSN: 1943-5886
  • URL: /core/journals/journal-of-symbolic-logic
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