Filters and ultrafilters over definable subsets of admissible ordinals

Abstract

The search for a recursive analogue of a measurable cardinal leads to a study of filters and ultrafilters over certain definable subsets of an admissible ordinal, using the hierarchy of constructible sets.

Connections with admissibility are explored in sections 2 and 3, and we find that the existence of a normal filter is stronger than the existence of the same type of filter (section 3). We look at the analogues of certain classical filters, namely the co-finite filter in section 2 and the normal filter of closed unbounded sets in section 3.

In section 4, we find that any filter (resp. normal filter) of a certain type can, on a countable ordinal, be extended to an ultrafilter (resp. normal ultrafilter) of the same type.

This article is part of the author's Ph.D. thesis, to appear under the same title at the University of Minnesota.