Is it possible in the $ZF$ theory to construct a function with domain $\omega_1$ (the set of all countable ordinals) that maps each countable ordinal $\alpha$ to a bijection between $\alpha$ and $\omega$?
I tried using some recursion and ended up constructing these up to $\epsilon_0$ (just then I found out about Cantor's normal form, this can indicate my knowledge about ordinals...). I figured that if I could construct for each countsble ordinal $\alpha$ some cofinal set for $\alpha$ that is not isomorphic to $\alpha$ itself then I could do this. Then I tried using the set of limit ordinals for that, repeating that step. I tried something else too later. But anyway, whatever I try I don't get further than $\epsilon_0$. That's probably because all my recursions revolve around basic ordinal operations: addition and multiplication. I am starting to doubt that it is possible.
It is consistent with $\mathsf{ZF}$ that no such function exists. Indeed, Feferman and Levy proved that it is consistent with $\mathsf{ZF}$ that $\omega_1$ is a countable union of countable sets (which obviously prevents such a function from existing, or perhaps more transparently would be prevented by the existence of such a function). Of course a proof of this is well beyond the scope of an MSE answer, but you can find relevant comments elsewhere on this site (e.g. here).
That said, it's easy to get well beyond $\epsilon_0$. Note that $\epsilon_0$ is a computable ordinal: there is a computable binary relation $R$ on $\mathbb{N}$ such that $(\mathbb{N};R)\cong\epsilon_0$. Via "pick the least index for a computable presentation," we can get an explicit bijection between $\mathbb{N}$ and the smallest noncomputable ordinal $\omega_1^{CK}$. We can with more thought continue much further. Incidentally, for a nice summary of ordinals beyond $\omega_1^{CK}$, I recommend this survey of Madore.
(And of course $\mathsf{ZF}$ does prove "For each countable ordinal $\alpha$ there is a function $f$ sending each $\beta<\alpha$ to an injection $\beta\rightarrow\omega$.")