In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets which are equinumerous). The concepts are developed by defining equinumerosity in terms of functions and the concepts of one-to-one and onto (injectivity and surjectivity); this gives us an pseudo-ordering relation

on the whole universe by size. It is not a true ordering because the trichotomy law need not hold: if both and , it is true by the Cantor-Bernstein-Schroeder theorem that A = _cB i.e. A and B are equinumerous, but they do not have to be literally equal; that at least one case holds turns out to be equivalent to the Axiom of choice.

Nevertheless, most of the interesting results on cardinality and its arithmetic can be expressed merely with =_c.

The goal of a cardinal assignment is to assign to every set A a specific, unique set which is only dependent on the cardinality of A. This is in accordance to Cantor's original vision of a cardinals: to take a set and abstract its elements into canonical "units" and collect these units into another set, such that the only thing special about this set is its size. These would be totally ordered by the relation and =_c would be true equality. As Y. N. Moschovakis says, however, this is mostly an exercise in mathematical elegance, and you don't gain much unless you are "allergic to subscripts." However, there are various valuable applications of "real" cardinal numbers in various models of set theory.

In modern set theory, we usually use the Von Neumann cardinal assignment which uses the theory of ordinal numbers and the full power of the Axioms of choice and replacement. Cardinal assignments do need the full Axiom of choice, if we want a decent cardinal arithmetic and an assignment for all sets. More on this (and much more good set theory in general!) can be found in Moschovakis' excellent introduction to set theory.

Reference

Moschovakis, Yiannis N. Notes on Set Theory. New York: Springer-Verlag, 1994.