Power set modulo small, the singular of uncountable cofinality

Let μ be singular of uncountable cofinality. If μ > 2cf(μ), we prove that in P = ([μ],⊇) as a forcing notion we have a natural complete embedding of Levy(א0, μ+) (so P collapses μ+ to א0) and even Levy(א0,UJbd κ (μ)). The “natural” means that the forcing ({p ∈ [μ] : p closed},⊇) is naturally embedded and is equivalent to the Levy algebra. Moreover we prove more than conjectured: if P fails the χ-c.c. then it collapses χ to א0. We even prove the parallel results for the case μ > א0 is regular or of countable cofinality. We also 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] for some 〈Aα : α < κ〉 ∈ P we have α < κ ⇒ |Aα ∩A| = κ. This research was supported by the United States-Israel Binational Science Foundation. Publication 861. I would like to thank Alice Leonhardt for the beautiful typing. Typeset by AMS-TEX 1