On the Mitchell and Rudin-Kiesler orderings of ultrafilters

The purpose of this paper is to study the consistency correlations between the Rudin-Kiesler and the Mitchell orderings of ultrafilters. There is no direct connection between these two orderings. Thus, in general, U 6nK V does not imply UCIV or converse. Moreover, if U cRK V and UaV, then there exists a p-measurable cardinal. It is easy to construct a Rudin-Kiesler increasing sequence of length o from one measurable cardinal, although it is not enough for the Mitchell increasing sequence even of length 2. But for a Rudin-Kiesler sequence of length w + 1 one measurable is not enough as was shown by Kanamori [5] and Mitchell [13]. On the other hand, Mitchell [13] constructed a Rudin-Kiesler sequence of length o2 from a Mitchell sequence of length 2. We are going to study the following equation: Con(a-increasing sequence of length x) iff Con( +,-increasing sequence of length y). Similar equations for Rudin-Kiesler sequences of Pand Q-points are also considered. Under the assumption “there is no inner model satisfying 3~ O(K) = K++" the solutions to these equation are given. It generalizes [13] which provides a partial solution for the Q-points case. Some related questions about the number of skies, constellations and existence of non-closed elementary embeddings will be considered. These notions were introduced and studied by Kanamori [5] and recently by Sureson [18]. It is worth noting that the Mitchell ordering or more precisely its rank function “0” is widely used as a scale for measuring the strength of propositions. So the above equations are actually considered in the form “Con+ O(K) = X) iff Con( +,-increasing sequence of length y)“. We are using some ideas and techniques of Mitchell [ll-141 and [l-3]. The paper is organized as follows. In Section 1 we find bounds on the length of Rudin-Kiesler increasing sequences of Q-points. These bounds depend only on the core model with the maximal sequence of ultrafilters. In Section 2 using the forcing arguments it is shown that the bounds of Section 1 are exact. The following theorem is proved.