Von Neumann cardinal assignment

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,

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 . This is a well-ordering of cardinal numbers.