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On the definability of the double jump in the computably enumerable sets
 J. MATH. LOG
, 2002
"... We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: Let C = {a: a is the Turing degree of a � 0 3 set J ≥T 0 ′ ′}. Let D ⊆ C such that D is upward closed in C. Then there is an L(A) property ϕD(A) such that F ′ ′ ∈ D iff there is an A where A ..."
Abstract

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We show that the double jump is definable in the computably enumerable sets. Our main result is as follows: Let C = {a: a is the Turing degree of a � 0 3 set J ≥T 0 ′ ′}. Let D ⊆ C such that D is upward closed in C. Then there is an L(A) property ϕD(A) such that F ′ ′ ∈ D iff there is an A where A ≡T F and ϕD(A). A corollary of this is that, for all n ≥ 2, the highn (lown) computably enumerable degrees are invariant in the computably enumerable sets. Our work resolves Martin’s Invariance Conjecture.