Descriptive properties on admissible sets

Abstract

Relations are treated between the following descriptive properties on admissible sets: enumerability, uniformization, reduction, separation, extension. Moreover, in the setting of these properties, we consider existence problems for a universal computable function and for a computable function universal for {0; 1}-valued computable functions. It is shown that all relations between the given properties are strict. Also we look into algorithmic complexity of admissible sets lending support to the specified relations. It is stated that the reduction principle fails in some admissible sets over classical structures.For the majority of approaches to computability, a class of computably enumerable sets, which, as a rule, form a lattice with respect to set-theoretic operations, is not closed under complements. (Moreover, elements that have complements are exactly computable sets.) This necessitates the study of additional set-theoretic properties of the given class. In the present paper, we look at such properties for admissible sets. It will be shown that admissible sets behave rather badly in relation to much used properties. Namely, every relation between the properties may be realized on a certain admissible set, and in most cases, such a set can be chosen as a hereditarily finite superstructure over a computable model.