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Power set modulo small, the singular of uncountable cofinality

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah*
Affiliation:
The Hebrew University of Jerusalem, Einstein Institute of Mathematics, Edmond J. Safra Campus, Givat Ram Jerusalem 91904, Israel Department of Mathematics, Hill Center-Busch Campus Rutgers, The State University of New Jersey, 110 Frelinghuysen Road Piscataway, NJ 08854-8019, USA. E-mail: shelah@math.huji.ac.ilURL: http://shelah.logic.at/

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 ⇒ ∣AαA∣ = κ.

Type
Research Article
Copyright
Copyright © Association for Symbolic Logic 2007

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