On inner constructivizability of admissible sets

We consider a problem of inner constructivizability of admissible sets by means of elements of a bounded rank. For hereditary finite superstructures we find the precise estimates for the rank of inner constructivizability: it is equal $\omega$ for superstructures over finite structures and less or equal 2 otherwise. We introduce examples of structures with hereditary finite superstructures with ranks 0, 1, 2. It is shown that hereditary finite superstructure over field of real numbers has rank 1.