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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:

03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E40 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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