A Limit on Relative Genericity in the Recursively Enumerable Sets

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 0' and high if X' computes 0". 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 E9 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 <P. Simplifying the situation, a typical strategy converts a failure to recursively compute W into a constraint on the enumeration of A, so that (W E9 A)' is forced to disagree with <P(-; 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 non-zero degree whose join with any other low degree is not high. 1