The class of cardinal numbers is well ordered

Question

I'm looking for a proof that the class of cardinal numbers is well ordered under the order relation $|A|\leq |B| \Leftrightarrow$ exists an injection $f:A \to B$.

In fact, I've found a very beautiful proof in an article, but I think it's not the standard one.

Answer 1

There are several standard arguments here:

1. We define the $\aleph$ by transfinite induction as a function from the ordinals. That means that there is a surjection (in fact bijection) between the cardinals and the ordinals. Therefore the aleph numbers are well-ordered.

2. We define $\aleph_\alpha$ to be the cardinality of the $\alpha$ initial ordinal. Therefore, we treat the alephs as ordinals. A subclass of the ordinals is always well-ordered.