On the existence of strong chains in ℘(ω1)/Fin

descriptionPublicationkeyboard_double_arrow_right Article 01 Sep 1998 English Publisher:Cambridge University Press (CUP)Journal:Journal of Symbolic Logic, volume 63, pages 1,055-1,062 (issn: 0022-4812, eissn: 1943-5886,

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Authors: Piotr Koszmider;

doi: 10.2307/2586726

On the existence of strong chains in ℘(ω1)/Fin

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Abstract

Abstract(Xα: α < ω2) ⊂ ℘(ω1) is a strong chain in ℘(ω1)/Fin if and only if Xβ – Xα is finite and Xα – Xβ is uncountable for each β < α < ω1. We show that it is consistent that a strong chain in ℘(ω1) exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in ℘(ω1) but no strong chain exists: is used to construct a c.c.c forcing that adds a strong chain and Chang's Conjecture to prove that there is no strong chain.

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