## 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:

 [1] Handbook of boolean algebras 2 pp 333–388– (1989) [2] Commentationes Mathematicae Universitatis Carolinae 29 pp 631–646– (1988) [3] Fundamenta Mathematical CX pp 11–24– (1980) [4] Proceedings of the American Mathematical Society 100 pp 205–212– (1987) [5] Israel Journal of Mathematics 92 pp 263–272– (1995) [6] Applications of PCF theory 65 pp 1624–1674– (2000) [7] Cardinal arithmetic 29 (1994) · Zbl 0848.03025 [8] Annals of Pure and Applied Logic 109 pp 117–129, math.LO/0009079– (2001) [9] Annals of Mathematical Logic 9 pp 401–439– (1976) [10] The mathematics of Paul Erdos, II 14 pp 420–459– (1997)
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