A simple $P_{{\aleph }_1}$-point and a simple $P_{{\aleph }_2}$-point

Abstract

We answer the long-standing question whether it is consistent to have simple $P$-points of two different characters. For a filter $\mathcal{F}$ over $\omega$ Guzmán and Kalajdzievski introduced a parametrised version of Miller forcing called $\mathbb{PT}(\mathcal{F})$. By combining iterands of the type $\mathbb{PT}(\mathcal{F})$ with others we establish: It is consistent relative to ZFC that there is a simple $P_{{\aleph }_1}$-point and a simple $P_{{\aleph }_2}$-point. A main technical point is the use of properness and descriptive complexity in the limit steps of uncountable cofinality.