# Axiom:Axiom of the Empty Set/Set Theory

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## 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

- Definition: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: $\text {ZF2}$