Von Neumann cardinal assignment - Definition

The von Neumann cardinal assignment is a cardinal assignment which uses ordinal numbers. For a well-ordered set U, we define its cardinal number to be the smallest ordinal number equinumerous to U. More precisely,

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

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. With the full Axiom of choice, 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 <math>\leq_c<math>. This is a well-ordering of cardinal numbers.

See also ordinal number, cardinal number, cardinal assignment.