Cardinal assignment

In set theory, the concept of cardinality is significantly developable without recourse to actually defining cardinal numbers as objects in the theory itself (this is in fact a viewpoint taken by Frege; Frege cardinals are basically equivalence classes on the entire universe of sets, by equinumerosity). 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 a quasi-ordering relation

${\displaystyle A\leq _{c}B\quad \iff \quad (\exists f)(f:A\to B\ \mathrm {is\ injective} )}$

on the whole universe by size. It is not a true partial ordering because antisymmetry need not hold: if both ${\displaystyle A\leq _{c}B}$ and ${\displaystyle B\leq _{c}A}$, it is true by the Cantor–Bernstein–Schroeder theorem that ${\displaystyle A=_{c}B}$ i.e. A and B are equinumerous, but they do not have to be literally equal (see isomorphism). That at least one of ${\displaystyle A\leq _{c}B}$ and ${\displaystyle B\leq _{c}A}$ 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 that is only dependent on the cardinality of A. This is in accordance with Cantor's original vision of 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 ${\displaystyle \leq _{c}}$, 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.