Positive undecidable numberings in the Ershov hierarchy

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.