No abstract

It was known that any non-single-element (in particular, any infinite) family of total functions with an oracle , such that , does not have -computable principal numbering; later it was proved that any finite family of total functions with a hyperimmune-free oracle always has an -computable principal numbering. The unresolved question was whether there exists an infinite family of total functions with a hyperimmune-free oracle that has an -computable principal numbering. The paper gives a positive answer to this question: it is proved that there exists an infinite -computable family of total functions, where the Turing degree of the set is hyperimmune-free, such that has an -computable principal numbering.