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arXiv 2012.023062020-12math.LO

The ultrafilter and almost disjointness numbers

Osvaldo Guzman, Damjan Kalajdzievski

英文摘要：

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 $ω_{1}=\mathfrak{u}<\mathfrak{a,}$ 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.

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