Embedding $\aleph_1$ into $\mathbb R$

Question

I am having trouble proving that one cannot construct an order-preserving map from $\aleph_1$ into $\mathbb R$.

I know that $\aleph_1$ equals the set of countable ordinals, but the fact that the map need not be surjective has made this trickier than I thought.

Is there a relatively elementary proof out there?

Answer 1

Let $\omega_1$ be the first uncountable ordinal. Assume it order-embedded in $\Bbb R$. Each $\alpha\in\omega_1$ has a successor $\alpha'$ in $\omega_1$. Between $\alpha$ and $\alpha'$ in $\Bbb R$ there is a rational number $c_\alpha$. All the $c_\alpha$ are distinct, and the rationals are countable....