We prove the following three theorems on the enumeration degrees of ∑20 sets. Theorem A: There exists a nonzero noncuppable ∑20 enumeration degree. Theorem B: Every nonzero Δ20enumeration degree is cuppable to 0′e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low Δ20 enumeration degree with the anticupping property.

We show that for every enumeration degree $a there exists an e-degree c such that $a \leq c , and all degrees b, with $c \leq b , are properly Σ 0 2.

We discuss a notion of forcing that characterizes enumeration 1-genericity, and we investigate the immunity, lowness, and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator Δ such that, for any A, the set ΔA is enumeration 1-generic and has the same jump complexity as A. We deduce from this and other recent results from the literature that not only does every degree a bound an enumeration 1-generic degree b such that a'=b', but also that, (...) if a is nonzero, then we can find such b satisfying 0eshrink)

Defining a class of sets to be uniform Δ02 if it is derived from a binary {0,1}{0,1}-valued function f≤TKf≤TK, we show that, for any C⊆DeC⊆De induced by such a class, there exists a high Δ02 degree c which is incomparable with every degree b ϵ Ce \ {0e, 0'e}. 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 Δ02 degree. (shrink)

We show that there exist downwards properly $\Sigma _{2}^{0}$ (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.

We show that there exist downwards properly $\Sigma _{2}^{0}$ (in fact noncuppable) e-degrees that are not high. We also show that every high e-degree bounds a noncuppable e-degree.

We show that every splitting of ${0}_{\mathrm{e}}^{\prime }$ in the local structure of the enumeration degrees, $$\mathcal{G}_{e} , contains at least one low-cuppable member. We apply this new structural property to show that the classes of all $\mathcal{K}$ -pairs in $\mathcal{G}_{e}$ , all downwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees and all upwards properly ${\mathrm{\Sigma }}_{2}^{0}$ enumeration degrees are first order definable in $\mathcal{G}_{e}$.

It is shown that there exists a low2 Harrington non-splitting base-that is, a low2 computably enumerable (c.e.) degree a such that for any c.e. degrees x, y, if $0' = x \vee y$ , then either $0' = x \vee a$ or $0' = y \vee a$ . Contrary to prior expectations, the standard Harrington non-splitting construction is incompatible with the $low_{2}-ness$ requirements to be satisfied, and the proof given involves new techniques with potentially wider application.