Editing Von Neumann cardinal assignment (section)

{{refimprove|date=February 2010}}
The '''[[John von Neumann|von Neumann]] cardinal assignment''' is a [[cardinal assignment]] that uses [[ordinal number]]s. For a [[well-order]]able set ''U'', we define its [[cardinal number]] to be the smallest ordinal number [[Equinumerosity|equinumerous]] to ''U'', using the von Neumann definition of an ordinal number. More precisely:

:<math>|U| = \mathrm{card}(U) = \inf \{ \alpha \in \mathrm{ON} \ |\ \alpha =_c U \},</math>

where ON is the [[class (set theory)|class]] of ordinals.  This ordinal is also called the '''initial ordinal''' of the cardinal.

That such an ordinal exists and is unique is guaranteed by the fact that ''U'' is well-orderable and that the class of ordinals is well-ordered, using the [[Axiom schema of replacement|axiom of replacement]]. With the full [[axiom of choice]], [[well-ordering theorem|every set is well-orderable]], so every set has a cardinal; we order the cardinals using the inherited ordering from the ordinal numbers. This is readily found to coincide with the ordering via ≤<sub>''c''</sub>. This is a well-ordering of cardinal numbers.