Computable Positive and Friedberg Numberings in Hyperarithmetic

We point out an existence criterion for positive computable total $$ {\Pi}_1^1 $$ -numberings of families of subsets of a given $$ {\Pi}_1^1 $$ -set. In particular, it is stated that the family of all $$ {\Pi}_1^1 $$ -sets has no positive computable total $$ {\Pi}_1^1 $$ -numberings. Also we obtain a criterion of existence for computable Friedberg $$ {\Sigma}_1^1 $$ -numberings of families of subsets of a given $$ {\Sigma}_1^1 $$ - set, the consequence of which is the absence of a computable Friedberg $$ {\Sigma}_1^1 $$ -numbering of the family of all $$ {\Sigma}_1^1 $$ -sets. Questions concerning the existence of negative computable $$ {\Pi}_1^1 $$ - and $$ {\Sigma}_1^1 $$ -numberings of the families mentioned are considered.