The ultrafilter and almost disjointness numbers

TL;DR

This paper proves the consistency of certain inequalities involving ultrafilter and almost disjointness numbers, demonstrating that MAD families can be destroyed without losing P-points, and resolves longstanding open problems in set theory.

Contribution

It introduces a proper forcing method that destroys MAD families while preserving P-points, and establishes the consistency of 1=<, solving longstanding open problems.

Findings

MAD families can be destroyed by proper forcing preserving P-points

It is consistent that 1=<

Provides a simple proof of the 1<s inequality consistency

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 $\mathfrak{u<s}$ is consistent.