## A model in which the base-matrix tree cannot have cofinal branches.(English)Zbl 0637.03049

Let $$[\omega]^{\omega}$$ be the set of all infinite subsets of $$\omega$$. $$x\subseteq_*y$$ means that x-y is finite. The distributivity cardinal $${\mathfrak h}$$ is the least $$\kappa$$ such that forcing with $$<[\omega]^{\omega},\subseteq_*>$$ adds a new function $$f: \kappa\to V$$. An $$\alpha$$-tower in $$[\omega]^{\omega}$$ is a descending $$\alpha$$- sequence $$<x_{\beta}: \beta <\alpha >$$ such that there is no $$x\in [\omega]^{\omega}$$ almost contained in all the $$x_{\beta}.$$
The author constructs a model of ZFC in which the distributivity cardinal $${\mathfrak h}$$ is $$2^{\aleph_ 0}=\aleph_ 2$$ and in which there are no $$\aleph_ 2$$-towers in $$[\omega]^{\omega}$$. This is done by iterating Mathias forcing. Mathias forcing is factorized into a countably closed and a $$\sigma$$-centered part. Then the author uses a mixture of finite and countable supports.
Reviewer: M.Weese

### MathOverflow Questions:

Relations between two tower numbers

### MSC:

 03E35 Consistency and independence results 03C62 Models of arithmetic and set theory
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