Suppes' Axiom of Cardinal Numbers

Question

In Suppes' book, Axiomatic Set Theory he introduces an axiom concerning cardinal numbers,before introducing them, namely that each set is associated with an object known as a cardinal number, and that if two sets have the same cardinal number then they are equipollent.

He states that later on he can obtain the cardinal numbers via another method, using something known as ordinal numbers and the axiom of choice, but he does not state if that will lead exactly to the axiom he's stated so far as I can tell. To anyone familiar with this book (1972) I'd like to ask the question - is there any point reading the section that uses this axiom? Will this axiom be entirely derived later on using the methods of ordinal numbers?

Answer 1

See Ch.8 : The Axiom of Choice; page 242 :

Theorem 8 : For any set $A$ there is a unique cardinal number $\alpha$ such that $\alpha \approx A$.

This is nothing other than the axiom for cardinal number [page 111] postulating that :

with each set $A$ is associated an object $\mathfrak K(A)$, the cardinal number of $A$, such that with two equipollent sets we associate the same cardinal number.