Abstract: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 G\"odel's Constructible Universe.
Keywords: tree, collectionwise Hausdorff, metrizable, Aronszajn tree
AMS Subject Classification: 54A35, 54E35, 54F05