TITLE: The regularity of successor cardinal
QUESTION [7 upvotes]: I was looking at two different proofs of the fact that successor cardinals are regular. It struck me as odd that both proofs used AC. Looking at the concepts involved in defining cofinality I feel as if there should a proof of this that doesn't use AC.
So, is my intuition correct or is this statement dependent on AC? Also in either case could you point me to a reference to the fact?

REPLY [7 votes]: Let me add on Andres' wonderful answer, something which I think is missing.
Despite the necessity of large cardinals, it was in fact one of the first uses of the technique of forcing (and symmetric extensions) to show that in fact $\omega_1$ and $\Bbb R$ can be countable unions of countable sets.
This is a classical result due to Feferman and Levy. In her Ph.D. Thesis Ioanna Dimitriou extended this result, and proved the following theorem:

Theorem (2.10) If $V$ is a model of $\sf ZFC$, $\kappa_0$ is a regular cardinal of $V$ and $\rho$ is an ordinal in $V$, then there is a model of $\sf ZF$ with a sequence of successive alternating singular and regular cardinals that starts at $\kappa_0$ and that contains $\rho$-many singular cardinals.

And all that without large cardinals. The large cardinals come into play when we want two successive singular cardinals.