First Uncountable Ordinal Cofinality: Needs AC? Say $\omega_1$ is the first uncountable ordinal. The reason I care about $\omega_1$ is


*

*Any countable subset of $\omega_1$ is bounded (or if you prefer, there is no countable cofinal subset).


This is obvious because


*A countable union of countable sets is countable.


Now, I'm told that we cannot prove (2) without at least some weak version of AC. Question: Does (1) also require AC?
Seems possible that some ordinalistic magic lets us be explicit about the relevant sequence of bijections?
 A: Yes, it requires the axiom of choice.
It was shown by Feferman and Levy, as one of the first uses of forcing, that it is consistent that $\omega_1$ is the countable union of countable sets. This means that there is a countable subset of $\omega_1$ which is unbounded. This is just assuming the consistency of $\sf ZFC$. (The idea is simple, for each $n$ add many bijections between $\omega$ and $\omega_n$; if you are careful enough, you can ensure that these are all the new bijections you add, and therefore there is no bijection between $\omega$ and the "original" $\omega_\omega$, which makes it the least uncountable ordinal of the model.)
This result was later generalized by Gitik who showed that assuming additional axioms are consistent, we can construct a model where every limit ordinal is the countable union of smaller ordinals. It was shown that at least some additional axioms are needed for such a result, although it seems that Gitik's original assumptions are probably stronger than needed.
