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March 2007 Power set modulo small, the singular of uncountable cofinality
Saharon Shelah
J. Symbolic Logic 72(1): 226-242 (March 2007). DOI: 10.2178/jsl/1174668393

Abstract

Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in ℛ = ([μ]μ,⊇) as a forcing notion we have a natural complete embedding of Levy(ℵ₀,μ⁺) (so ℛ collapses μ⁺ to ℵ₀) and even Levy(ℵ₀,UJbdκ(μ)). The “natural” means that the forcing ({p ∈ [μ]μ:p closed},⊇) is naturally embedded and is equivalent to the Levy algebra. Also if ℛ 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 𝔟κ partitions Ā=〈 Aα:α<κ〉 of κ such that for any A∈ [κ]κ for some 〈 Aα:α<κ〉 ∈ P we have α<κ→ |Aα ∩ A|=κ.

Citation

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Saharon Shelah. "Power set modulo small, the singular of uncountable cofinality." J. Symbolic Logic 72 (1) 226 - 242, March 2007. https://doi.org/10.2178/jsl/1174668393

Information

Published: March 2007
First available in Project Euclid: 23 March 2007

zbMATH: 1116.03039
MathSciNet: MR2298480
Digital Object Identifier: 10.2178/jsl/1174668393

Rights: Copyright © 2007 Association for Symbolic Logic

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Vol.72 • No. 1 • March 2007
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