Axiom:Axiom of the Empty Set

Axiom

The Axiom of the Empty Set posits the existence of a set which has no elements.

Depending on whether this axiom is declared in the context of set theory or class theory, it exists in different forms.

Set Theory

$\exists x: \forall y: \paren {\neg \paren {y \in x} }$

Class Theory

In class theory, the existence of the empty class is not axiomatic, as it has been derived from previous axioms.

Hence the Axiom of the Empty Set takes this form:

The empty class $\O$ is a set, that is:

$\O \in V$

where $V$ denotes the basic universe.

Also known as

In the specific context of set theory, the Axiom of the Empty Set is also known as the axiom of existence, but there exists another axiom with such a name, used in a different context.

Hence it is preferable not to use that name.

Some sources refer to this as the axiom of the null set.

Also see

• Equivalence of Formulations of Axiom of Empty Set for Classes

• Definition:Zermelo-Fraenkel Axioms
• Definition:Empty Set
• Empty Set is Unique
• If Set Exists then Empty Set Exists