This entity is de ned to be the union of all stages, which consist of all sets with the same rank and form the cumulative   hierarchy.
If omega is taken to be the set of natural numbers, then this entity sub omega is a model for set theory without the axiom of in nity, while this entity sub omega plus omega is a model for Zermelo set theory. This entity is constructed by trans nite induction over the ordinal numbers. This entity is de ned to be the union of all stages, which consist of all sets with the same rank and form the cumulative   hierarchy. For ten points, identify this entity symbolized V , the class of hereditary and well-founded sets and foundation for modern set theories.
If omega is taken to be the set of natural numbers, then this entity sub omega is a model for set theory without the axiom of in nity, while this entity sub omega plus omega is a model for Zermelo set theory.
von Neumann universe [accept von Neumann hierarchy of sets before  hierarchy  and prompt afterward, prompt throughout on just hierarchy of sets]