= Rationally labelled Aronszajn tree construction
{title2=$\ell:T\to\mathbb Q,\ s<_Tt\Longrightarrow\ell(s)<\ell(t)$}
Construct countable levels of bounded increasing rational sequences, maintaining the <bounded rational extension property>. At successor stages append every allowed rational. At a countable limit, handle each earlier-node/rational-cap pair by a cofinal sequence of extensions with values approaching a rational strictly below the cap. Its union has that rational supremum. Countably many requirements give a countable new level. Strictly increasing labels exclude uncountable chains, and label fibers are <tree antichains>, so the result is a <special Aronszajn tree>.
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