## Abstract

A sufficient condition is given under which an infinite computable family of Σ ^{-1} _{a} -sets has computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved for finite levels of the Ershov hierarchy in [1]. As a consequence, it is stated that the family of all Σ ^{-1} _{a} -sets has a computable positive undecidable numbering. In addition, for every ordinal notation a > 1, an infinite family of Σ ^{-1} _{a}-sets is constructed which possesses a computable positive numbering but has no computable Friedberg numberings. This answers the question of whether such families exist at any-finite or infinite-level of the Ershov hierarchy, which was originally raised by Badaev and Goncharov only for the finite levels bigger than 1.

Original language | English |
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Pages (from-to) | 512-525 |

Number of pages | 14 |

Journal | Algebra and Logic |

Volume | 50 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jan 2012 |

## Keywords

- Ershov hierarchy
- positive undecidable numbering

## ASJC Scopus subject areas

- Analysis
- Logic