Rationally labelled Aronszajn tree construction ($\ell :T\to \mathbb{Q},s{<}_{T}t⟹\ell (s)<\ell (t)$)

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.

For every earlier node and rational cap above its label, an extension at each later level remains below the cap. In the rationally labelled Aronszajn tree construction this reserves space for limit-stage unions. Choosing an intermediate rational and forcing successive values toward it ensures a rational supremum and strictly increasing labels even at limit levels.