Nevertheless, most of the *interesting* results on cardinality and its arithmetic can be expresed 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.

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