P-points and Q-points over a measurable cardinal

Abstract

This paper presents constructions of κ-ultrafilters over a measurable cardinal κ, specifically p-points and q-points, in the continuation of works of Gitik, Kanamori and Ketonen. On the assumption of supercompactness, Kunen has shown the existence of a chain of p-points, in the Rudin–Keisler ordering, of maximal length. We shall prove a similar result for q-points. Results of Mitchell [8,9] establish connections between Rudin—Keisler chains of κ-ultrafilters and inner models of “∃ ν( o( ν)= ν ++)”. This shows the necessity of some strong large cardinal hypothesis. The second part of the paper is devoted to a separation property of κ- ultrafilters (cf. Kanamori and Taylor). To answer a question of Taylor concerning the existence of a non-separating p-point, we use a combination of Silver's Forcing and iterated ultrapowers; the proof itself may be of some interest.