Absolute and Relative Properties of Negatively Numbered Families

We study in this paper negative 𝔸 -numberings where 𝔸 are admissible structures. We establish that a series of classical assumptions is remained to hold for negative 𝔸 -numberings in the case of the application of certain limits as for numberings as for admissible structures. We find examples of admissible structures 𝔸 whose families of all Σ -subsets have negative non-decidable minimal computable 𝔸 -numberings. The admissible structures from this series have negative computable 𝔸 -numberings whose numbered equivalence differs from corresponding ones of any computable 𝔸 -numbering of family of total functions.