Journal of Symbolic Logic
- J. Symbolic Logic
- Volume 54, Issue 1 (1989), 160-176.
Complementation in the Turing Degrees
Posner  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.
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. https://projecteuclid.org/euclid.jsl/1183742858