Computable numberings of families of infinite sets

We state the following results: the family of all infinite computably enumerable sets has no computable numbering; the family of all infinite $\Pi^{1}_{1}$ sets has no $\Pi^{1}_{1}$-computable numbering; the family of all infinite $\Sigma^{1}_{2}$ sets has no $\Sigma^{1}_{2}$-computable numbering. For $k>2$, the existence of a $\Sigma^{1}_{k}$-computable numbering for the family of all infinite $\Sigma^{1}_{k}$ sets leads to the inconsistency of $ZF$.