# 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.

See also ordinal number, cardinal number, cardinal assignment.