Definable Ultrapowers and Ultrafilters over Admissible Ordinals

Q 0. Introduction and summary of results The present paper develops the theory of definable ultrafilters over constructible sets. Section 1 is devoted to preliminaries and notation (where one can find, in particular, definitions of the terms used in this introduction). Although the paper is selfcontained, some knowledge of the ideas in KAUFMA" [4] and KRANAKIS [6] might be helpful; the main results are reviewed in Section 1. In Section 2 we develop the basic properties of the ultrapowers A ( P ) and Ult(F). The main result of that section is: Theorem 2.9. The following are equivalent for any ITn-uf F on x : (i) Ult(F) r A(F); (ii) F is A,-based. In Section 3 we study the &collection schema in ultrapowers, e.g. Corol lary 3.10, 3.8. If F is a IIn-nuf, and n = 1 or [n 2 2 and L, 1 i (power set