There may be simple $\text{Pℵ}{}_1$ and $\text{Pℵ}{}_2\text{-}\text{points}$ and the Rudin-Keisler ordering may be downward directed

Abstract

We prove the consistency, relative to ZFC, of each of the following two (mutually contradictory) statements. (A) Every two non-principal ultrafilters on m have a common image via a finite-to-one function. (B) Simple $\text{Pℵ}{}_1\text{-}\text{points}$ and simple $\text{Pℵ}{}_2\text{-}\text{points}$ both exist. These results, proved by the second author, answer questions of the first author and P. Nyikos, who had obtained numerous consequences of (A) and (B), respectively. In the models we construct, the bounding number is $\text{ℵ}{}_1$, while the dominating number, the splitting number, and the cardinality of the continuum are $\text{ℵ}{}_2$.