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:
“If you are convinced of a matter, you must take sides or you don’t deserve to succeed.”
—Johann Wolfgang Von Goethe (1749–1832)
“What a lesson here for our world. One blast, thousands of years of civilization wiped out.”
—Kurt Neumann (1906–1958)
“To this war of every man against every man, this also is consequent; that nothing can be Unjust. The notions of Right and Wrong, Justice and Injustice have there no place. Where there is no common Power, there is no Law; where no Law, no Injustice. Force, and Fraud, are in war the two Cardinal virtues.”
—Thomas Hobbes (1579–1688)