Positive undecidable numberings in the Ershov hierarchy

Abstract

We give a su cient condition for an in nite computable family of ����1 a sets, to have computable positive but undecidable numberings, where a is a notation for a nonzero computable ordinal. This extends a theorem proved by Talasbaeva for the nite levels of the Ershov hierarchy. In par- ticular the family of all ����1 a sets has positive undecidable numberings: this veri es for all levels of the Ershov hierarchy a conjecture due to Badaev and Goncharov. We point out also that for every ordinal notation a of a nonzero ordinal, there are families of ����1 a sets having positive numberings, but no Friedberg numberings: this answers for all levels (whether nite or in nite) of the Ershov hierarchy, a question originally raised, only for the nite levels over level 1, by Badaev and Goncharov.