Abstract

Let A be an infinite $\Delta _{2}^{0}$ 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 Wf(x) ∩ Wf(y) = ∅, if x ≠ y.) Using this result we prove that A is hyperhyperimmune if and only if no $\Delta _{2}^{0}$ subset B of A is s-complete. i.e., there is no $\Delta _{2}^{0}$ subset B of A such that K̄ ≤s B, where ≤s denotes s-reducibility, and K̄ denotes the complement of K