Title: On Inner Constructivizability of Admissible Sets

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

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