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.

中文翻译：

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超滤器和几乎不相交的数字

我们证明了每个MAD家族都可以通过适当的强制来破坏P点。有了这个结果，我们证明它是一致的${\omega }_1=\mathfrak{你}<\mathfrak{一种},$解决了一个近 20 年前的 Shelah 问题和一个 Brendle 问题。我们还将提供 Blass 和 Shelah 结果的一个简单证明，即不等式$\mathfrak{你}<\mathfrak{秒}$ 是一致的。