Reducibility on families

A reducibility on families of subsets of natural numbers is introduced which allows the family per se to be treated without its representation by natural numbers being fixed. This reducibility is used to study a series of problems both in classical computability and on admissible sets: for example, describing index sets of families belonging to $\Sigma_3^0 $, generalizing Friedberg's completeness theorem for a suitable reducibility on admissible sets, etc.