= Coherent-injection Aronszajn tree
{title2=$T=\{e_\alpha\upharpoonright\beta:\beta\le\alpha<\omega_1\}$}
Given <coherent coinfinite injections into omega>, use all restrictions $e_\alpha\upharpoonright\beta$, where $\beta\le\alpha<\omega_1$, ordered by proper extension. Every level is countable because its nodes differ finitely from a fixed <function> on a countable domain. An uncountable <chain in a partial order> would have unbounded domain heights, and its union would inject $\omega_1$ into $\omega$, which is impossible.
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