Von Neumann Universe

The von Neumann universe, or von Neumann hierarchy of sets, denoted V, is the class of hereditary well-founded sets studied in set theory; formalized by Zermelo-Fraenkel set theory, it is often used to interpret or motivate that theory's own axioms. It is named after John von Neumann, though it was first published by Ernst Zermelo in 1930. Each well-founded set is assigned a rank, defined inductively as the smallest ordinal greater than the ranks of all of its members, and the sets making up V are organized into a transfinite, cumulative hierarchy indexed by that rank.