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:
- ,
where ON is the 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 of replacement. 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 ≤c. This is a well-ordering of cardinal numbers.
Read more about Von Neumann Cardinal Assignment: Initial Ordinal of A Cardinal
Famous quotes containing the words von, neumann and/or cardinal:
“The force of a language does not consist of rejecting what is foreign but of swallowing it.”
—Johann Wolfgang Von Goethe (1749–1832)
“It means there are times when a mere scientist has gone as far as he can. When he must pause and observe respectfully while something infinitely greater assumes control.”
—Kurt Neumann (1906–1958)
“The Cardinal is at his wit’s end—it is true that he had not far to go.”
—George Gordon Noel Byron (1788–1824)