In set theory and related branches of mathematics, the von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets. This collection, which is formalized by Zermelo–Fraenkel set theory (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC.
The rank of a well-founded set is defined inductively as the smallest ordinal number greater than the ranks of all members of the set. In particular, the rank of the empty set is zero, and every ordinal has a rank equal to itself. The sets in V are divided into a transfinite hierarchy, called the cumulative hierarchy, based on their rank.
Read more about Von Neumann Universe: Definition, V and Set Theory, Philosophical Perspectives
Famous quotes containing the words von, neumann and/or universe:
“The true poet is called to take in the splendor of the world and for that reason will always be inclined to praise rather than to find fault.”
—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)
“Pray. To ask the laws of the universe to be annulled on behalf of a single petitioner confessedly unworthy.”
—Ambrose Bierce (1842–1914)