The ultrafilter and almost disjointness numbers

Osvaldo Guzman, Damjan Kalajdzievski

Published 2020-12-03Version 1

We prove that every \textsf{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 $\mathfrak{u<s}$ is consistent.