Von Neumann Universe - V and Set Theory

V and Set Theory

If ω is the set of natural numbers, then V_ω is the set of hereditarily finite sets, which is a model of set theory without the axiom of infinity. V_ω+ω is the universe of "ordinary mathematics", and is a model of Zermelo set theory. If κ is an inaccessible cardinal, then V_κ is a model of Zermelo-Fraenkel set theory (ZFC) itself, and V_κ+1 is a model of Morse–Kelley set theory.

V is not "the set of all sets" for two reasons. First, it is not a set; although each individual stage V_α is a set, their union V is a proper class. Second, the sets in V are only the well-founded sets. The axiom of foundation (or regularity) demands that every set is well founded and hence in V, and thus in ZFC every set is in V. But other axiom systems may omit the axiom of foundation or replace it by a strong negation (for example is Aczel's anti-foundation axiom). These non-well-founded set theories are not commonly employed, but are still possible to study.