Computable Positive and Friedberg Numberings in Hyperarithmetic

Abstract

© 2020, Springer Science+Business Media, LLC, part of Springer Nature. We point out an existence criterion for positive computable total Π11 -numberings of families of subsets of a given Π11 -set. In particular, it is stated that the family of all Π11 -sets has no positive computable total Π11 -numberings. Also we obtain a criterion of existence for computable Friedberg Σ11 -numberings of families of subsets of a given Σ11 - set, the consequence of which is the absence of a computable Friedberg Σ11 -numbering of the family of all Σ11 -sets. Questions concerning the existence of negative computable Π11 - and Σ11 -numberings of the families mentioned are considered.