The directedness of the Rudin-Keisler order at measurable cardinals

TL;DR

This paper explores the structure of ultrafilters on measurable cardinals, establishing a connection between the Rudin-Keisler order's directedness and the existence of certain large cardinals, with implications for set theory's compactness properties.

Contribution

It proves the equiconsistency of the Rudin-Keisler order's directedness with the existence of measurable cardinals having specific Mitchell order, linking combinatorial properties to large cardinal hypotheses.

Findings

The Rudin-Keisler order is $oxed{ ext{equiconsistent}}$ with measurable cardinals of certain Mitchell order.

The $ ext{Gluing Property}$ relates to the directedness of ultrafilter orders.

The $ ext{Gluing Property}$ fails in Gitik's classical model, affecting the structure of ultrafilters.

Abstract

The manuscript is concerned with the Rudin-Keisler order of ultrafilters on measurable cardinals. The main theorem proved read as follows: Given regular cardinals $\lambda\leq \kappa$, the following theories are equiconsistent modulo ZFC: (1) $\kappa$ is a measurable cardinal with $o(\kappa)=\lambda^+$ (resp. $o(\kappa)=\kappa$). (2) The Rudin-Keisler order restricted to the set of $\kappa$-complete (non-principal) ultrafilters on $\kappa$ is $\lambda^+$-directed (resp. $\kappa^+$-directed). The theorem reported here is proved after bridging the directedness of the RK-order with the $\lambda$-Gluing Property introduced by the authors in \cite{HP}. Our result provides what seems to be the first example of a compactness-type property at the level of measurable cardinals whose consistency strength is much lower than the existence of a strong cardinal. As part of our analysis we also answer a question of Gitik by showing that the $\aleph_0$-Gluing Property fails in his classical model from ''Changing cofinalities and the nonstationary ideal". As a consequence of this, in Gitik's model the Rudin-Keisler order fails to be $\aleph_1$-directed.