Journal of Symbolic Logic

On the Existence of Strong Chains in $\wp(\omega_1)$/Fin

Piotr Koszmider

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Abstract

$(X_\alpha : \alpha < \omega_2) \subset \wp(\omega_1)$ is a strong chain in $\wp(\omega_1)$/Fin if and only if $X_\beta - X_\alpha$ is finite and $X_\alpha - X_\beta$ is uncountable for each $\beta < \alpha < \omega_1$. We show that it is consistent that a strong chain in $\wp(\omega_1)$ exists. On the other hand we show that it is consistent that there is a strongly almost-disjoint family in $\wp(\omega_1)$ but no strong chain exists: $\square_{\omega_1}$ 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.

Article information

Source
J. Symbolic Logic, Volume 63, Issue 3 (1998), 1055-1062.

Dates
First available in Project Euclid: 6 July 2007

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1183745580

Mathematical Reviews number (MathSciNet)
MR1649075

Zentralblatt MATH identifier
0936.03043

JSTOR
links.jstor.org

Citation

Koszmider, Piotr. On the Existence of Strong Chains in $\wp(\omega_1)$/Fin. J. Symbolic Logic 63 (1998), no. 3, 1055--1062. https://projecteuclid.org/euclid.jsl/1183745580


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