Jump equivalence of the Δ20 hyperimmune sets

An infinite set A is said to be hyperhyperimmune (h.h.i.) if, for any collection of disjoint simultaneously recursively enumerable (r.e.) finite sets, A must fail to intersect with one of those sets. Thus the elements of an h.h.i. set are, in a sense, very elusive. D. A. Martin [3] showed that the degrees of h.h.i. sets with r.e. complements are exactly the r.e. degrees with jump 0″ . More generally, C. G. Jockusch [2] found a′ ≥ 0″ to be a sufficient condition for a to be the degree of an h.h.i. set and found a′ ≥ 0′ to be necessary. However, it was also shown that in the degrees as a whole neither condition gave a characterization of the h.h.i. degrees. The purpose of this note is to prove that a′ = 0″ does characterize the h.h.i. degrees below 0′ . Theorem. The degrees below 0′ containing h.h.i. sets are exactly those degrees below 0′ with jump 0″ . Proof. From [2], if a′ ≥ 0″ , then a contains an h.h.i. set. Conversely, let A ∈ a where a′ 0″ and a 0′ . Let { A s ∣ s ≥ 0} be a recursive sequence of finite sets such that for each x , lim s , A x ( x ) exists and equals A ( x ). For a set B , let B [ m ] denote B ∩ [0, m ], and (if B is finite) let ∣ B ∣ denote the cardinality of B .