Positive Numberings of Families of Sets in the Ershov Hierarchy

It is proved that there exist infinitely many positive undecidable $\Sigma^{-1}_n$-computable numberings of every infinite family $\mathcal S\subseteq\Sigma^{-1}_n$ that admits at least one $\Sigma^{-1}_n$-computable numbering and contains either the empty set, for even $n$, or $N$ for odd $n$.