Computability in special models

Abstract

We study some properties of descriptive set theory which translate from the ideals of enumerability degrees to admissible sets. We show that the reduction principle fails in the admissible sets corresponding to nonprincipal ideals and possessing the minimality property and that the properties of existence of a universal function, separation, and total extension translate from the ideals to some special classes of admissible sets. We first give some examples of the admissible sets satisfying the total extension principle. In addition, we define a broad subclass of admissible sets admitting no decidable computable numberings of the family of all computably enumerable subsets. We mostly discuss the minimal classes of admissible sets corresponding to the nonprincipal ideals of enumerability degrees.