Problem from Kunen II.40:
The definition is the following: An $\omega_1$-Aronszajn tree $T$ called special iff $T$ is the union of $\omega$ antichains.
Need to prove that $T$ is special iff there is a map $f: T \rightarrow \mathbb{Q}$ such that for $x,y \in T, x < y \rightarrow f(x) < f(y)$, and show that a special Aronszajn tree exist.
The hint is to construct $T$ and $f$ simultaneously by induction.
It seems like I should somehow "pack" the antichains (equivalence classes?), to achieve a map from a "large" tree to a relatively small set. Couldn't advance any furher than that, though.
Any help?
Thanks in advance.