Regularity of $\omega_1$ and axiom of choice

Question

Why is the regularity of the ordinal $\omega_1$ a consequence of the axiom of choice?

Answer 1

We can prove, without the axiom of choice, the following theorem:

If $\langle A_i\mid i\in\Bbb N\rangle$ is a sequence of sets, and $\langle f_i\mid i\in\Bbb N\rangle$ is a sequence of injections $f_i\colon A_i\to\Bbb N$, then $\bigcup_{i\in\Bbb N}A_i$ is countable.

Namely, if the enumerations of each $A_i$ were given, then we can enumerate the union. The axiom of choice is needed to uniformly enumerate all the sets, if the enumerations are not given. Namely, by the definition of countable we know that $F_i=\{f\mid f\colon A_i\to\Bbb N\text{ injective}\}$ is a non-empty set; and using the axiom of choice we can choose $f_i\in F_i$ for all $i\in\Bbb N$.

So it follows from the axiom of choice that countable unions of countable sets are countable. Assuming choice, if so, given any countable collection of countable ordinals, their union is a countable ordinal again. So $\omega_1$ is regular.

But it turns out that the axiom of choice, or at least a weak form of it, is really needed to run this proof through. One of the first applications of forcing was to show that the axiom of choice can fail, and soon after Feferman and Levy showed it is consistent that the axiom of choice fails, and $\omega_1$ is the countable union of countable ordinals.