## The club principle and the distributivity number.(English)Zbl 1231.03040

Ostaszewski’s club principle $$\clubsuit$$ is the statement that there is a sequence of sets $$(A_\alpha : \alpha < \omega _1$$ is a limit ordinal$$)$$ such that all $$A_\alpha$$ are cofinal in $$\alpha$$ and every uncountable $$X \subseteq \omega_1$$ contains some $$A_\alpha$$. $$\clubsuit$$ is a weakening of Jensen’s classical diamond $$\diamondsuit$$, which is consistent with the negation of the continuum hypothesis. Let $${\mathfrak h}$$ denote the distributivity number of $${\mathcal P} (\omega) / {\mathrm{fin}}$$, $${\mathfrak g}$$ the groupwise density number, and $${\mathfrak u}$$ the ultrafilter number.
The author proves that $$\clubsuit$$ is consistent with $${\mathfrak h} = \aleph_2$$ by showing that $$\clubsuit$$ holds in the iterated Mathias model. This answers a question of Hrušák and the reviewer. For the proof, she singles out a finiteness property of an Axiom A forcing notion $${\mathbb P}$$, which basically says that, for all $$n$$, $$q \leq_n p$$ is determined by $$q \leq p$$ and a finite piece of $$p$$, where $$\leq_n$$ denotes the $$n$$-th ordering of $${\mathbb P}$$. Next she shows that, if $$\diamondsuit$$ holds in the ground model, then $$\clubsuit$$ holds in the extension by an $$\omega_2$$-stage iteration of Axiom A forcings with the finiteness property. As a further application, the consistency of $${\mathfrak u} < {\mathfrak g}$$ with $$\clubsuit$$ is mentioned.

### MSC:

 3e+35 Consistency and independence results 300000 Other combinatorial set theory 3e+15 Descriptive set theory 3e+17 Cardinal characteristics of the continuum
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### References:

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