Von Neumann cardinal assignment
From Wikipedia, the free encyclopedia.
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.
