Von Neumann universe

The class of all sets arranged in a cumulative hierarchy, where each stage contains every subset of the preceding stage. Its stages are indexed by ordinals.

Ordinal number: An ordinal is an order type of a well-ordered set, used to index positions in transfinite sequences. Ordinals name the stages at which the cumulative hierarchy grows.
Vα: The set at hierarchy level α, defined from earlier levels by the cumulative construction. Each Vα is one indexed stage of the universe.
Von Neumann–Bernays–Gödel set theory: A set theory that treats classes as objects while distinguishing sets from proper classes. It formalizes the universe as a proper class alongside its sets.
Axiom of Foundation: The axiom asserting that every nonempty set has an element disjoint from it. Foundation ensures membership descends through the hierarchy without infinite regress.
Power set: The power set of a set is the set of all its subsets. Taking power sets generates successor stages of the hierarchy.
Successor ordinal: An ordinal of the form α + 1, immediately following an ordinal α. At successor stages, the next level is the power set of the current one.
Constructible universe: The inner model L, containing sets obtainable by a definability-based construction. Unlike the full hierarchy, it retains only constructible sets.
Axiom of Power Set: The axiom asserting that every set has a set containing exactly its subsets. It guarantees each successor stage exists as a set.
Transfinite recursion: A method for defining objects in stages indexed by ordinals, using earlier stages at successor and limit steps. It specifies the hierarchy at successor and limit ordinals.
Limit ordinal: A nonzero ordinal that is not a successor ordinal. At limit stages, the hierarchy takes the union of all earlier levels.

Linked from 5 pages