My question is: let's say that a tree T has no cofinal branch and has all levels of size $< \mu$. Can we prove (in ZFC) that the set of branches of T has cardinality $\leq \mu$? I have a very strong intuition that it is so, but I can't come up with a proof (or a counterexample).
2026-03-27 03:58:09.1774583889
Tree without cofinal branches and with levels of bounded size
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No.
Suppose that $T$ is a normal Aronszajn tree (so any point has splitting successors) and $2^{\aleph_0}>\aleph_1$. Then there are no cofinal branches, but easily due to splitting at any successor point, any limit stage will have continuum many branches, of which only countably many are allowed to be realised and extended.