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Complementation in the Turing degrees

  • Theodore A. Slaman (a1) and John R. Steel (a2)

Posner [6] has shown, by a nonuniform proof, that every degree has a complement below 0′. We show that a 1-generic complement for each set of degree between 0 and 0′ can be found uniformly. Moreover, the methods just as easily can be used to produce a complement whose jump has the degree of any real recursively enumerable in and above ∅′. In the second half of the paper, we show that the complementation of the degrees below 0′ does not extend to all recursively enumerable degrees. Namely, there is a pair of recursively enumerable degrees a above b such that no degree strictly below a joins b above a. (This result is independently due to S. B. Cooper.) We end with some open problems.

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Ά#x005B;2Ά#x005D; C. G. Jockusch and R. A. Shore , REA operators, r. e. degrees and minimal covers, Recursion theory, Proceedings of Symposia in Pure Mathematics, vol. 42, American Mathematical Society, Providence, Rhode Island, 1985, pp. 3Ά#x2013;11.

Ά#x005B;3Ά#x005D; S. C. Kleene and E. L. Post , The upper semi-lattice of degrees of recursive unsolvability, Annals of Mathematics, ser. 2, vol. 59 (1954), pp. 379Ά#x2013;407.

Ά#x005B;11Ά#x005D; C. Spector , On degrees of recursive unsolvability, Annals of Mathematics, ser. 2, vol. 64 (1956), pp. 581Ά#x2013;592.

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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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