Limit lemmas and jump inversion in the enumeration degrees

Abstract.We show that there is a limit lemma for enumeration reducibility to 0e', analogous to the Shoenfield Limit Lemma in the Turing degrees, which relativises for total enumeration degrees. Using this and `good approximations' we prove a jump inversion result: for any set W with a good approximation and any set X<eW such that W≤eX' there is a set A such that X≤eA<eW and A'=W'. (All jumps are enumeration degree jumps.) The degrees of sets with good approximations include the Σ02 degrees and the n-CEA degrees.