Draw a regular $n$-gon on the unit circle. Rotate it so that its vertices have distinct horizontal coordinates. This is possible because each pair of vertices excludes only finitely many angles.

Sort the numbers as $c_1<\cdots<c_n$ and place them from left to right in this order. Write $w(P)$ for the label at vertex $P$. Thus

$$x(P)<x(Q)\quad\Longleftrightarrow\quad w(P)<w(Q).$$

We will show that this labeling works for every length.