Power set modulo small, the singular of uncountable cofinality

Abstract

Let μ be singular of uncountable cofinality. If μ > 2^cf(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,U_J^bdκ(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 𝔟_κ partitions Ā=〈 A_α:α<κ〉 of κ such that for any A∈ [κ]^κ for some 〈 A_α:α<κ〉 ∈ P we have α<κ→ |A_α ∩ A|=κ.