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 pseudo-ordering relation

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.

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