Journal of Symbolic Logic

Power set modulo small, the singular of uncountable cofinality

Saharon Shelah

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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.

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J. Symbolic Logic, Volume 72, Issue 1 (2007), 226-242.

First available in Project Euclid: 23 March 2007

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Shelah, Saharon. Power set modulo small, the singular of uncountable cofinality. J. Symbolic Logic 72 (2007), no. 1, 226--242. doi:10.2178/jsl/1174668393.

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