The ultrafilter and almost disjointness numbers

Abstract

We prove that every MAD family can be destroyed by a proper forcing that preserves $P$-points. With this result, we prove that it is consistent that ${\omega }_1=\mathfrak{u}<\mathfrak{a},$ solving a nearly 20 year old problem of Shelah and a problem of Brendle. We will also present a simple proof of a result of Blass and Shelah that the inequality $\mathfrak{u}<\mathfrak{s}$ is consistent.