Power set modulo small, the singular of uncountable cofinality.(English)Zbl 1116.03039

Summary: Let $$\mu$$ be singular of uncountable cofinality. If $$\mu> 2^{\text{cf} (\mu)}$$, we prove that in $$\mathbb{P}=([\mu]^\mu,\supseteq)$$ as a forcing notion we have a natural complete embedding of Levy$$(\aleph_0,\mu^+)$$ (so $$\mathbb{P}$$ collapses $$\mu^+$$ to $$\aleph_0)$$ and even Levy$$(\aleph_0,{\mathbf U}_{J_\kappa^{\text{bd}}}(\mu))$$. The “natural” means that the forcing $$(\{p\in[\mu]^\mu:p$$ closed},$$\supseteq)$$ is naturally embedded and is equivalent to the Levy algebra. Also if $$\mathbb{P}$$ fails the $$\chi$$-c.c. then it collapses $$\chi$$ to $$\aleph_0$$ (and the parallel results for the case $$\mu>\aleph_0$$ is regular or of countable cofinality). Moreover we prove: for regular uncountable $$\kappa$$, there is a family $${\mathbf P}$$ of $${\mathfrak b}_\kappa$$ partitions $$\widetilde A=\langle A_\alpha:\alpha<\kappa\rangle$$ of $$\kappa$$ such that for any $$A\in[\kappa ]^\kappa$$ for some $$\langle A_\alpha:\alpha< \kappa\rangle\in{\mathbf P}$$ we have $$\alpha<\kappa\Rightarrow |A_\alpha\cap A|=\kappa$$.

MSC:

 30000 Ordered sets and their cofinalities; pcf theory 300000 Other combinatorial set theory 3e+40 Other aspects of forcing and Boolean-valued models
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References:

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