Turing Degrees of Hyperjumps
Abstract
The Posner-Robinson Theorem states that for any reals Z and A such that Z 0' ≤T A and 0 <T Z, there exists B such that A T B' T B Z T B 0'. Consequently, any nonzero Turing degree degT(Z) is a Turing jump relative to some B. Here we prove the hyperarithmetical analog, based on an unpublished proof of Slaman, namely that for any reals Z and A such that Z O ≤T A and 0 <HYP Z, there exists B such that A T OB T B Z T B O. As an analogous consequence, any nonhyperarithmetical Turing degree degT(Z) is a hyperjump relative to some B.
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