ON THE DEFINABILITY OF THE DOUBLE JUMP IN THE COMPUTABLY ENUMERABLE SETS
Peter A. Cholak, Leo A. Harrington · Journal of Mathematical Logic · 2002
We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: let [Formula: see text] is the Turing degree of a [Formula: see text] set J ≥T0″}. Let [Formula: see text] such that [Formula: see text] is upward closed in [Formula: see text]. Then there is an ℒ(A) property [Formula: see text] such that [Formula: see text] if and only if there is an A where A ≡T F and [Formula: see text]. A corollary of this is that, for all n ≥ 2, the high n ([Formula: see text]) computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin's Invariance Conjecture.