Axiom:Axiom of the Empty Set/Set Theory

Axiom

There exists a set that has no elements:

Formulation 1

$\exists x: \forall y \in x: y \ne y$

Formulation 2

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

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

• Axiom:Zermelo-Fraenkel Axioms
• Definition:Empty Set
• Empty Set is Unique
• Axiom of Empty Set from Axiom of Infinity and Axiom of Specification

Sources

• 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (previous) ... (next): $\S 4$: Set Theory: $4.2$ The Zermelo-Fraenkel axioms: $\text {ZF2}$