Von Neumann universe

In <a href="/facts/Set_theory/1ptfsvUP">set theory</a> and related branches of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the von Neumann universe, or von Neumann hierarchy of sets, denoted by V, is the <a href="/facts/Class_(set_theory)/F5EmDd9b">class</a> of <a href="/facts/Hereditary_set/edFwmXa8">hereditary</a> <a href="/facts/Well-founded_set/zyRRyJ4M">well-founded sets</a>. This collection, which is formalized by <a href="/facts/Zermelo%E2%80%93Fraenkel_set_theory/UYIsPbXw">Zermelo–Fraenkel set theory</a> (ZFC), is often used to provide an interpretation or motivation of the axioms of ZFC. The concept is named after <a href="/facts/John_von_Neumann/aUk6urof">John von Neumann</a>, although it was first published by <a href="/facts/Ernst_Zermelo/uvW8gDq0">Ernst Zermelo</a> in 1930.
The rank of a well-founded set is defined inductively as the smallest <a href="/facts/Ordinal_number/GelFLjJt">ordinal number</a> greater than the ranks of all members of the set. In particular, the rank of the <a href="/facts/Empty_set/ffEk3eA6">empty set</a> is zero, and every ordinal has a rank equal to itself. The sets in V are divided into the <a href="/facts/Transfinite_induction/XlolgTDl">transfinite</a> hierarchy Vα , called the cumulative hierarchy, based on their rank.

Von Neumann universe open-in-new

Von Neumann universe