On the generability of admissible sets.

*(Russian)*Zbl 0648.03027The author defines the notion of the smallest admissible set generated by a structure contained in an admissible set. It generalizes the notion of the admissible cover of a model of set theory given by J. Barwise [Admissible sets and structures. An approach to definability theory (1975; Zbl 0316.02047)]. The author points out an example of a structure contained in an admissible set such that it does not generate an admissible set. Let M be a structure contained in an admissible set A such that the elements of M are pairwise disjoint subsets of the class of all urelements of A. The first theorem of the paper states that there is the smallest admissible set generated by M and the order type of the ordinals in it is \(\omega\). A generalization of Theorem 2.2 from the Appendix of the cited book is proved. Some applications of the theorems and their proofs to \(\Sigma\)-admissible sets and \(\Sigma\)-predicates are obtained.

Reviewer: A.N.Ryaskin

##### MSC:

03C70 | Logic on admissible sets |

03D60 | Computability and recursion theory on ordinals, admissible sets, etc. |