 Rationally labelled Aronszajn tree construction

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.