Journal of Symbolic Logic

A characterization of the Δ⁰₂ hyperhyperimmune sets

Roland Sh. Omanadze and Andrea Sorbi

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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.

Article information

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

First available in Project Euclid: 27 December 2008

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

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

Q-reducibility s-reducibility hyperhyperimmune set


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.

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