## Journal of Symbolic Logic

- J. Symbolic Logic
- Volume 73, Issue 4 (2008), 1407-1415.

### A characterization of the Δ⁰₂ hyperhyperimmune sets

Roland Sh. Omanadze and Andrea Sorbi

#### 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.

#### Article information

**Source**

J. Symbolic Logic, Volume 73, Issue 4 (2008), 1407-1415.

**Dates**

First available in Project Euclid: 27 December 2008

**Permanent link to this document**

https://projecteuclid.org/euclid.jsl/1230396928

**Digital Object Identifier**

doi:10.2178/jsl/1230396928

**Mathematical Reviews number (MathSciNet)**

MR2467226

**Zentralblatt MATH identifier**

1161.03026

**Subjects**

Primary: 03D30: Other degrees and reducibilities 03D25: Recursively (computably) enumerable sets and degrees

**Keywords**

Q-reducibility s-reducibility hyperhyperimmune set

#### Citation

Omanadze, Roland Sh.; Sorbi, Andrea. A characterization of the Δ⁰₂ hyperhyperimmune sets. J. Symbolic Logic 73 (2008), no. 4, 1407--1415. doi:10.2178/jsl/1230396928. https://projecteuclid.org/euclid.jsl/1230396928