Possible inaccuracy in Wikipedia article about initial ordinals I quote from the Wikipedia article:
"So (assuming the axiom of choice) we identify $\omega_\alpha$  with $\aleph_\alpha$, except that the notation $\aleph_\alpha$ is used for writing cardinals, and $\omega_\alpha$ for writing ordinals. "
Let's remind ourselves of the definitions: 
(Def.1) The $\aleph_\alpha$ numbers are defined recursively, with $\aleph_0 = |\mathbb N| = \omega = \omega_0$ and $\aleph_{\alpha + 1} =$ the least ordinal such that it has strictly greater cardinality than $\aleph_\alpha$ for $\alpha \in \mathbf{ON}$.
(Def.2) The initial ordinal of a cardinal $\kappa$ is defined to be the least ordinal of cardinality $\kappa$. Then $\omega_\alpha$ are the transfinite initial ordinals.
Then by definition, every $\aleph$ is an ordinal and also by definition, $\aleph_\alpha = \omega_\alpha$, with or without AC. What am I missing? Thanks for clarification. 
 A: Perhaps they're alluding to the fact that without the axiom of choice, not all infinite cardinals are alephs; with it, the finite ordinals and the $\aleph_\alpha$ take care of all the cardinals.
A: The $\aleph$ numbers are defined as the initial ordinals, without any appeal to the axiom of choice. This is done by a transfinite recursion over the ordinals. $\aleph_\alpha$ is therefore the ordinal $\omega_\alpha$, which is the unique initial ordinal such that the set of all initial ordinals strictly smaller than itself has order type $\alpha$.
In some places define cardinals only as $\aleph$ numbers, and then you need AC to assert that every set has a cardinal number. Of course you can do it without AC but then there are non-$\aleph$ cardinals as well.

One should note that before Dana Scott suggested the definition of a cardinal without the axiom of choice much later than von Neumann suggested using the $\aleph$ numbers. Historically, if so, it was "obvious" how to assign cardinals when the axiom of choice was present, but not without it. 
