Zermelo–Fraenkel set theory

Zermelo–Fraenkel set theory is an axiomatic foundation for mathematics in which every mathematical object is represented as a set and sets obey a specified collection of axioms.

Axiom of Extensionality: Sets with exactly the same elements are equal. It fixes when two sets count as the same object.
Set-theoretic encoding: A method of representing mathematical structures and objects using sets and membership. It explains how numbers, functions, and structures can all be treated as sets.
Zermelo–Fraenkel set theory with the axiom of choice: Zermelo–Fraenkel set theory supplemented by the axiom of choice. This extension, commonly called ZFC, is the standard alternative when choice is included.
Natural number: A nonnegative integer used for counting and ordering. The Infinity axiom supports constructing the natural numbers within the theory.
Axiom Schema of Separation: For any set and property, the elements satisfying that property form a subset. It permits restricted subsets without allowing unrestricted comprehension.
Von Neumann ordinal: An ordinal represented as the set of all smaller ordinals. This construction represents each natural number as the set of its predecessors.
Axiom of Choice: An axiom asserting that a choice function exists for every family of nonempty sets. ZF does not include this principle, whose addition changes what the theory can prove.
Ordinal number: A number that represents the order type of a well-ordered set. Ordinals extend the set-theoretic construction of natural numbers to transfinite orders.
Axiom of Pairing: For any two sets, a set exists whose elements are precisely those two sets. It supplies basic finite sets from which larger constructions can proceed.
Ordered pair: A set-theoretic construction that represents two objects in a specified order. Pairing and other set operations encode ordered relationships without primitive tuples.

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