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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:

03E35 Consistency and independence results
03E05 Other combinatorial set theory
03E15 Descriptive set theory
03E17 Cardinal characteristics of the continuum
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