On Non-principal Arithmetical Numberings and Families

Abstract

The paper studies \(\varvec{\Sigma ^0_n}\)-computable families (\(\varvec{n\geqslant 2}\)) and their numberings. It is proved that any non-trivial \(\varvec{\Sigma ^0_n}\)-computable family has a complete with respect to any of its elements \(\varvec{\Sigma ^0_n}\)-computable non-principal numbering. It is established that if a \(\varvec{\Sigma ^0_n}\)-computable family is not principal, then any of its \(\varvec{\Sigma ^0_n}\)-computable numberings has a minimal cover and, if the family is infinite, is incomparable with one of its minimal \(\varvec{\Sigma ^0_n}\)-computable numberings. It is also shown that for any \(\varvec{\Sigma ^0_n}\)-computable numbering \(\varvec{\nu }\) of a \(\varvec{\Sigma ^0_n}\)-computable non-principal family there exists its \(\varvec{\Sigma ^0_n}\)-computable numbering that is incomparable with \(\varvec{\nu }\). If a non-trivial \(\varvec{\Sigma ^0_n}\)-computable family contains the least and greatest elements under inclusion, then for any of its \(\varvec{\Sigma ^0_n}\)-computable non-principal non-least numberings \(\varvec{\nu }\) there exists a \(\varvec{\Sigma ^0_n}\)-computable numbering of the family incomparable with \(\varvec{\nu }\). In particular, this is true for the family of all \(\varvec{\Sigma ^0_n}\)-sets and for the families consisting of two inclusion-comparable \(\varvec{\Sigma ^0_n}\)-sets (semilattices of the \(\varvec{\Sigma ^0_n}\)-computable numberings of such families are isomorphic to the semilattice of \(\varvec{m}\)-degrees of \(\varvec{\Sigma ^0_n}\)-sets).