How far is it possible to develop cardinals without ordinals? I'm wondering which of the usual facts about cardinals in ZFC can be established without using ordinal arithmetic at all. After all the definitions of a cardinal (as a class of equivalence), and also of limit/successor/regular/singular cardinals etc. do not involve order types. Is it possible to develop "just" cardinals, without working with ordinals? Is there an analogue of transfinite recursion that "just" uses cardinals? Does there exist a treatment of cardinals written up along these lines?
 A: Well, how far is a bit difficult to answer. Because one can rarely know if what they know on the topic is "as far as it can get".
But you can develop some of the basic theory. All the cardinal arithmetic is doable. You can even prove that $a^2=a$ using Zorn's lemma without appealing directly to ordinals. And this will allow you to even develop basic cardinal exponentiation consequences in $\sf ZFC$.
You can even develop the notion of cofinality can be defined even without ordinals, but using partitions and cardinals instead, and talk about Koenig's theorem.
But I don't think that you can get truly far. You can still define what is a successor cardinal, and what are limit cardinals, but without the ordinals to index them, you're missing quite a bit. The deep structure of the cardinals (which is the little they have left after the axiom of choice trivializes the arithmetic basics and nothing substantial can be proved about exponentiation besides Koenig's theorem).
So the question is, how far are you trying to get? If you want something which is in the confines of a basic course in elementary set theory, sure you can do all that without talking about ordinals, but you're missing out. Because ordinal make the structure of the cardinals, and you'll be touching the trunk of an elephant instead of taking ten steps back and admiring the beast.
And as for written references. I know of none, because set theory books usually aim to teach ordinals as well. For what it's worth, if you're willing to pick and choose, you can find proofs for all of the above on this very site.
