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A minimal pair of recursively enumerable degrees

  • C. E. M. Yates (a1)
Abstract

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.

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[1] R. M. Friedberg , Two recursively enumerable sets of incomparable degrees of unsolvability, Proceedings of National Academy of Science, U.S.A., vol. 43 (1957), pp. 236238.

[4] E. L. Post , Recursively enumerable sets of positive integers and their decision-problems, Bulletin of the American Mathematical Society, vol. 50 (1944), pp. 284316.

[6] G. E. Sacks , The recursively enumerable degrees are dense, Annals of Mathematics, vol. 80 (1964), pp. 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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