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

Published online by Cambridge University Press:  12 March 2014

Piotr Koszmider
Piotr Koszmider*
Affiliation:
Department of Mathematics, Auburn University, Auburn, AL 36849, USA Departamento de Matemática, Universidade de São Paulo, Caixa Postal: 66281, São Paulo, S.P. CEP: 05315-970, Brasil E-mail: piotr@ime.usp.br
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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.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 63 , Issue 3 , September 1998 , pp. 1055 - 1062
Copyright
Copyright © Association for Symbolic Logic 1998

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