Editing Cardinal assignment (section)

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 [[Gottlob Frege|Frege]]; Frege cardinals are basically [[equivalence class]]es on the entire [[universe (mathematics)|universe]] of [[Set (mathematics)|sets]], by [[equinumerous|equinumerosity]]). The concepts are developed by defining equinumerosity in terms of functions and the concepts of [[injective function|one-to-one]] and [[surjective function|onto]] (injectivity and surjectivity); this gives us a [[preorder|quasi-ordering]] relation

:<math>A \leq_c B\quad \iff\quad (\exists f)(f : A \to B\ \mathrm{is\ injective})</math>

on the whole universe by size. It is not a true [[partial ordering]] because [[antisymmetric relation|antisymmetry]] need not hold: if both <math>A \leq_c B</math> and <math>B \leq_c A</math>, it is true by the [[Cantor–Bernstein–Schroeder theorem]] that <math>A =_c B</math> i.e. ''A'' and ''B'' are equinumerous, but they do not have to be literally equal (see [[isomorphism]]). That at least one of <math>A \leq_c B</math> and <math>B \leq_c A</math> holds turns out to be equivalent to the [[axiom of choice]].

Nevertheless, most of the ''interesting'' results on cardinality and its [[cardinal arithmetic|arithmetic]] can be expressed merely with =<sub>c</sub>.

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 [[Georg Cantor|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 <math>\leq_c</math>, and =<sub>c</sub> would be true equality. As [[Yiannis N. Moschovakis|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 [[model theory|models]] of set theory.

In modern set theory, we usually use the [[Von Neumann cardinal assignment]], which uses the theory of [[ordinal number]]s and the full power of the axioms of [[Axiom of choice|choice]] and [[Axiom of replacement|replacement]]. Cardinal assignments do need the full axiom of choice, if we want a decent cardinal arithmetic and an assignment for ''all'' sets.