A characterization of the Δ⁰₂ hyperhyperimmune sets

Abstract

Let A be an infinite Δ₂⁰ set and let K be creative: we show that K≤_Q A if and only if K≤_Q₁ A. (Here ≤_Q denotes Q-reducibility, and ≤_Q₁ is the subreducibility of ≤_Q obtained by requesting that Q-reducibility be provided by a computable function f such that W_f(x)∩ W_f(y)=∅, if x \not= y.) Using this result we prove that A is hyperhyperimmune if and only if no Δ⁰₂ subset B of A is s-complete, i.e., there is no Δ⁰₂ subset B of A such that \overline{K}≤_s B, where ≤_s denotes s-reducibility, and \overline{K} denotes the complement of K.