Here's the problem as stated:

Show if $\displaystyle \alpha$ is any countable ordinal, then $\displaystyle \exists f:\alpha \rightarrow \Re$ (to the reals) where $\displaystyle f$ is order preserving... So $\displaystyle \beta \epsilon \gamma \epsilon \alpha \Rightarrow f(\beta) < f(\alpha) $.

Now, the only idea I have is to create a function that will map any countable ordinal to $\displaystyle \Re$, for instance:

$\displaystyle 1 \rightarrow .1$

$\displaystyle 2 \rightarrow .12 $

$\displaystyle 3 \rightarrow .123 $

$\displaystyle : $

$\displaystyle \omega \rightarrow .12345... $

So this cover through $\displaystyle \omega$. Do I need it to cover through $\displaystyle \omega_{1}$, the first uncountable ordinal? If so, what is an example of something that can get that large? Maybe all the irrationals, mapped similarly as above? Any help would be appreciated.

Note: I'm trying to avoid using the continuum hypothesis.