Journal of Symbolic Logic

Complementation in the Turing Degrees

Theodore A. Slaman and John R. Steel

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Posner [6] has shown, by a nonuniform proof, that every $\triangle^0_2$ degree has a complement below 0'. We show that a 1-generic complement for each $\triangle^0_2$ 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 $\varnothing'$. 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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J. Symbolic Logic, Volume 54, Issue 1 (1989), 160-176.

First available in Project Euclid: 6 July 2007

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Slaman, Theodore A.; Steel, John R. Complementation in the Turing Degrees. J. Symbolic Logic 54 (1989), no. 1, 160--176.

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