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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 ..."
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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.
On The Filter Of Computably Enumerable Supersets Of An RMaximal Set
 Arch. Math. Logic
, 2001
"... . We study the filter L (A) of computably enumerable supersets (modulo finite sets) of an rmaximal set A and show that, for some such set A, the property of being cofinite in L (A) is still \Sigma 0 3 complete. This implies that for this A, there is no uniformly computably enumerable " ..."
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. We study the filter L (A) of computably enumerable supersets (modulo finite sets) of an rmaximal set A and show that, for some such set A, the property of being cofinite in L (A) is still \Sigma 0 3 complete. This implies that for this A, there is no uniformly computably enumerable "tower" of sets exhausting exactly the coinfinite sets in L (A). 1. The theorem The computably enumerable (or recursively enumerable) sets form a countable sublattice (denoted by E in the following) of the power set P(!) of the set of natural numbers. The operations of union and intersection are effective on E (i.e., effective in the indices of the computably enumerable sets). The complemented elements of E are exactly the computable sets. The finite sets in E are definable as those elements only bounding complemented elements; thus studying E is closely related to studying E , the quotient of E modulo the ideal of finite sets. (From now on, the superscript will always denote that we are wor...