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 \(\alpha<\kappa\Rightarrow |A_\alpha\cap A|=\kappa\).


03E04 Ordered sets and their cofinalities; pcf theory
03E05 Other combinatorial set theory
03E40 Other aspects of forcing and Boolean-valued models
Full Text: DOI arXiv


[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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.