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 ω1 = u < 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 u < s is consistent.