I'm revising Kunen's Set Theory and he mentions that when we define the cardinality of a set, we should do it with well-orderable sets. Why is this the case? He points to a section which contains many statements equivalent to AC and the proofs of their equivalence. I know that AC is equivalent to every set being well-orderable, but I don't see why we would have to have $A$ well-orderable in order to define it's cardinality. Naively, one could say it's the least ordinal which is in bijective correspondence with the set $A$.
Any help would be appreciated!
Well, if you “define” the cardinality of a set to be the least ordinal that bijects with it, then for the definition to be well defined you need to show that there exists such an ordinal, but the existence of such an ordinal is equivalent to the well orderability of the set since you can just transport the well ordered structure from the ordinal to the set.
Note that it is still possible to define cardinality for non well orderable sets (in the absence of choice) using Scott’s Trick.