# Article

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Keywords:
tree; collectionwise Hausdorff; metrizable; Aronszajn tree
Summary:
It is independent of the usual (ZFC) axioms of set theory whether every collectionwise Hausdorff tree is either metrizable or has an uncountable chain. We show that even if we add ``or has an Aronszajn subtree,'' the statement remains ZFC-independent. This is done by constructing a tree as in the title, using the set-theoretic hypothesis \$\diamondsuit^*\$, which holds in Gödel's Constructible Universe.
References:
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