Numberings on Admissible Structures over Equivalence Relations

The paper gives a series of examples of admissible sets in which the family of all -c.e. sets has neither negative, nor positive computable -numberings. In particular, it is proved that for hereditarily finite superstructures over negative partners of equivalence relations the family of all -subsets may have no negative computable numbering. An opposite result was established earlier for positive numberings and positive partners of equivalence relations [ 2 , 4 ].