Limit lemmas and jump inversion in the enumeration degrees

Badness and jump inversion in the enumeration degrees

“…Griffith shows in [9] Griffith went on to show that any set in a total enumeration degree is jump uniform under ≤ e . However it turns out that this property applies to a larger class of sets.…”

“…Griffith also proved the existence of a set X that is not jump uniform under ≤ e (see Proposition 3.2 of [9]), and hence that, by Lemma 3.2, also is not good approximable. The purpose of this section is to construct, using the underlying ideas of Griffith's proof, a 0 2 set B that is not jump uniform under ≤ e and such that B (which is ≤ e J ∅ ) is low 2 -where we note that this implies that B itself is such that J B ≤ e J (2) ∅ .…”

“…The first tools for inverting the jump in the enumeration degrees were developed by Griffith in [9]. In effect, Griffith observed that, if any sets A and B have the property P(A, B) which states that A consists precisely of the indices of the finite columns of some set X enumeration reducible (≤ e ) to B-i.e.…”

“…The results in [9] were extended in [10] where, in particular, it was shown that, for any good approximable set A the set { e | A e is infinite } is enumeration equivalent to J (2) A . This made available tools used later in [13] and [12] for constructing enumeration degrees with specific relative jump complexities.…”

“…Section 5 is dedicated to showing that this is the case. To do so it applies the methodology of [9] in a priority tree construction of a 0 2 set A such that J A ≡ e J (2) ∅ and whose enumeration degree is quasiminimal.…”