Minimal Generalized Computable Numberings and Families of Positive Preorders

Abstract

We study A-computable numberings for various natural classes of sets. For an arbitrary oracle A≥T0′, an example of an A-computable family S is constructed in which each A-computable numbering of S has a minimal cover, and at the same time, S does not satisfy the sufficient conditions for the existence of minimal covers specified in [Sib. Math. J., 43, No. 4, 616-622 (2002)]. It is proved that the family of all positive linear preorders has an A-computable numbering iff A′≥T0". We obtain a series of results on minimal A-computable numberings, in particular, Friedberg numberings and positive undecidable numberings.