Consider $\omega^2$. We can build a simple bijection between the ordinal and $\omega$ similarly to how the bijection between $\mathbb{Q}$ and $\mathbb{N}$ can be built.

I was wondering if there is a way to get such a bijection between all countable transfinite ordinals and $\omega$. In the title I used the word `canonical`

. Perhaps a way of defining this is: can we build a function $F$ which, given a transfinite ordinal $\tau$, returns a bijection $f$ with $\omega$, without using the axiom of choice?

I'm also satisfied if this is true only for recursive ordinals.

Thank you.