Avoiding uniformity in the Δ20 enumeration degrees

Defining a class of sets to be uniform Δ20 if it is derived from a binary {0,1}-valued function f≤TK, we show that, for any C⊆De induced by such a class, there exists a high Δ20 degree c which is incomparable with every degree b∈C∖{0e,0e′}. We show how this result can be applied to quite general subclasses of the Ershov Hierarchy and we also prove, as a direct corollary, that every nonzero low degree caps with both a high and a nonzero low Δ20 degree.