Well-ordering of sets of cardinal numbers 
Proposition For every cardinal number $m$ there is a definite next larger cardinal number.

This proposition is proved on page 136 of "Proofs from the Book" using the fact that any set of ordinal numbers is well-ordered. However the latter fact is presented without proof.
The reasoning looks kind of strange to me because it seems that we are proving the well-orderedness of any set of cardinal numbers by means of the same property for sets of ordinal numbers. (And I have a feeling that the proof for ordinals should be even harder than that of cardinals, but I must be wrong!)
I don't have any background in set theory or logic, but I was hoping someone could either direct me to a "non-technical" reference or perhaps share some insights on this. Thanks!

I would recommend one of two books for you to learn about the foundations of set theory,
Proofs and Fundamentals by Ethan D. Bloch
Topology by James B. Munkres
