There may be simple \(P_{\aleph _ 1}\)- and \(P_{\aleph _ 2}\)-points and the Rudin-Keisler ordering may be downward directed (Q1096622)

The consistency relative to ZFC of the following mutually contradictory statements is proven. (A) Every two non-principal ultrafilters on \(\omega\) have a common image under some finite-to-one function. (B) Simple \(P_{\aleph_ 1}\)-points and simple \(P_{\aleph_ 2}\)-points both exist. The consistency of (A) is established by an iterated forcing construction that makes up the bulk of the paper. Interestingly this same construction can be modified to show (B).