Work in the setting of the recursively enumerable sets and their Turing degrees. A set X is low if X ′, its Turing jump, is recursive in ∅′ and high if X ′ computes ∅″. Attempting to find a property between being low and being recursive, Bickford and Mills produced the following definition. W is deep , if for each recursively enumerable set A , the jump of A ⊕ W is recursive in the jump of A . We prove that there are no deep degrees other than the recursive one. Given a set W , we enumerate a set A and approximate its jump. The construction of A is governed by strategies, indexed by the Turing functionals Φ . Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A , so that ( W ⊕ A )′ is forced to disagree with Φ (−; A ′). The conversion has some ambiguity; in particular, A cannot be found uniformly from W . We also show that there is a “moderately” deep degree: There is a low nonzero degree whose join with any other low degree is not high.
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