Axiom of Empty Set

The axiom of empty set asserts that at least one set has no elements. In standard set theory, this set is unique and is denoted ∅.

Axiom: A statement accepted as a starting assumption within a formal system. The axiom of empty set is one assumption in an axiomatic theory.
Existential quantification: A logical quantifier asserting that at least one object satisfies a stated condition. The axiom uses an existence claim for a set with no members.
Ernst Zermelo: A German mathematician who formulated an influential axiomatic set theory in the early twentieth century. His axiomatization helped establish the framework in which the axiom is stated.
Natural number: A number used for counting or ordering, with conventions about whether zero is included. In the von Neumann construction, zero is identified with the empty set.
Set: A collection treated as a single mathematical object, governed by specified membership rules. The axiom asserts existence of an object that qualifies as a set.
Extensionality axiom: The set-theoretic axiom that sets with the same members are equal. It makes the empty set unique once existence is established.
Abraham Fraenkel: A mathematician whose work helped shape the Zermelo–Fraenkel axioms. The theory named partly for Fraenkel includes the empty-set existence principle.
Von Neumann ordinal: An ordinal represented as the set of all smaller ordinals. The empty set serves as the least ordinal in this construction.
Set membership: The relation stating that an object is an element of a set, written ∈. Having no elements means nothing bears the membership relation to this set.
Axiom schema of specification: A Zermelo–Fraenkel axiom schema allowing subsets defined by formulas within an existing set. Together with another set’s existence, it can produce an empty subset.

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Axiom of pairingCompared with: It guarantees a set with no members, unlike pairing's prescribed members.