### A characterization of the Δ⁰₂ hyperhyperimmune sets

Roland Sh. Omanadze and Andrea Sorbi
Source: J. Symbolic Logic Volume 73, Issue 4 (2008), 1407-1415.

#### 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 Wf(x)∩ Wf(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.

First Page:
Primary Subjects: 03D30, 03D25
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Permanent link to this document: http://projecteuclid.org/euclid.jsl/1230396928
Digital Object Identifier: doi:10.2178/jsl/1230396928
Mathematical Reviews number (MathSciNet): MR2467226
Zentralblatt MATH identifier: 1161.03026

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