About the admissible predicates on admissible sets

We study the admissible predicates, i.e., the predicates having the property that their addition to the signature of an admissible set preserves the property “to be an admissible set”. We show that the family of these predicates is much wider than the family of $\Delta$-predicates. We also construct a family of admissible predicates of cardinality $2^{\omega}$ such that the addition of an arbitrary pair of predicates of this family to the signature of an admissible set violates the admissibility of the latter as well as other examples of families of admissible predicates.