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 ($(\aleph _{0},U_{J_{\kappa}^{{\rm bd}}}(\mu))$). The "natural" means that the forcing ({p ∈ [μ]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 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