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Power set modulo small, the singular of uncountable cofinality

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road Piscataway, NJ 08854-8019, USA. E-mail:


Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in ℙ = ([μ]μ, ⊇) as a forcing notion we have a natural complete embedding of Levy(ℵ0, μ+) (so ℙ collapses μ+ to ℵ0) and even Levy (). The “natural” means that the forcing ({p ∈ [μ] : p closed}, ⊇) is naturally embedded and is equivalent to the Levy algebra. Also if ℙ fails the χ-c.c. then it collapses χ to ℵ0 (and the parallel results for the case μ > ℵ0 is regular or of countable cofinality). Moreover we prove: for regular uncountable κ, there is a family P of κ partitions Ā = ⟨Aα : α < κ⟩ of κ such that for any A ∈ [κ]κ for some ⟨Aα : α < κ⟩ ∈ P we have α < κ ⇒ ∣AαA∣ = κ.

Research Article
Copyright © Association for Symbolic Logic 2007

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[BaFr87] Balcar, Bohuslav and Franěk, František, Completion of factor algebras of ideals, Proceedings of the American Mathematical Society, vol. 100 (1987), pp. 205–212.CrossRefGoogle Scholar
[BPS] Balcar, Bohuslav, Pelant, Jan, and Simon, Petr, The space of ultrafilters on N covered by nowhere dense sets, Fundamenta Mathematical vol. CX (1980), pp. 11–24.Google Scholar
[BaSi88] Balcar, Bohuslav and Simon, Petr, On collections of almost disjoint families, Commentationes Mathematicae Universitatis Carolinae, vol. 29 (1988), pp. 631–646.Google Scholar
[BaSi89] Balcar, Bohuslav and Simon, Petr, Disjoint refinement, Handbook of boolean algebras, vol. 2, North-Holland, 1989, pp. 333–388.Google Scholar
[BaSi95] Balcar, Bohuslav and Simon, Petr, Baire number of the spaces of uniform ultrafilters, Israel Journal of Mathematics, vol. 92 (1995), pp. 263–272.CrossRefGoogle Scholar
[Ba] Baumgartner, James E., Almost disjoint sets, the dense set problem and partition calculus, Annals of Mathematical Logic, vol. 9 (1976), pp. 401–439.CrossRefGoogle Scholar
[KjSh:720] Kojman, Menachem and Shelah, Saharon, Fallen cardinals, Annals of Pure and Applied Logic, vol. 109 (2001), pp. 117–129, math.LO/0009079.CrossRefGoogle Scholar
[Sh:g] Shelah, Saharon, Cardinal arithmetic, Oxford Logic Guides, vol. 29, Oxford University Press, 1994.CrossRefGoogle Scholar
[Sh:506] Shelah, Saharon, The pcf-theorem revisited, The mathematics of Paul Erdős, II, Algorithms and Combinatorics, vol. 14, Springer, 1997, math.LO/9502233, pp. 420–459.Google Scholar
[Sh:589] Shelah, Saharon, Applications of PCF theory, this Journal, vol. 65 (2000), pp. 1624–1674.Google Scholar