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Power set modulo small, the singular of uncountable cofinality
Published online by Cambridge University Press: 12 March 2014
Abstract
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 ⇒ ∣Aα ∩ A∣ = κ.
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- Research Article
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- Copyright © Association for Symbolic Logic 2007
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