Cumulative hierarchy

The cumulative hierarchy is a sequence of set-theoretic levels built by iterating the power-set operation along the ordinals. Each level contains all sets appearing at earlier levels, and its union gives the universe of sets.

Power set: The power set of a set is the set of all its subsets. At each successor stage, taking a power set generates the next level.
Set: A set is a collection of distinct objects treated as a single mathematical object. The hierarchy organizes sets into levels rather than treating them as an unstructured totality.
Universe of sets: The universe of sets is the totality of sets considered in set theory, commonly denoted V. The cumulative hierarchy is the standard staged description of this universe.
Non-well-founded set theory: Non-well-founded set theory permits sets whose membership relation contains cycles or infinite descending chains. Such sets do not arise in the ordinary cumulative hierarchy.
Ernst Zermelo: Ernst Zermelo was a German mathematician who formulated an influential axiomatic set theory. His foundational work helped establish the framework in which staged set construction is studied.
Ordinal number: An ordinal is a transitive set well-ordered by membership, representing the order type of a well-order. Ordinals index the stages, including successor steps and limit stages.
Membership relation: The membership relation, written ∈, states that one object is an element of a set. Membership determines which sets must appear at earlier levels than a given set.
Constructible universe: The constructible universe L is an inner model built from sets definable over earlier stages. It adapts the hierarchy by admitting only sets definable at each stage.
Aczel's anti-foundation axiom: Aczel's anti-foundation axiom replaces foundation with a principle allowing certain circular membership structures. It provides a formal alternative to the hierarchy's well-founded construction.
John von Neumann: John von Neumann was a Hungarian-American mathematician who developed the ordinal-based approach to set theory. His work connected ordinal stages with a systematic universe of sets.

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