# Axiom:Axiom of Empty Set/Set Theory

Jump to navigation
Jump to search

## Contents

## Axiom

In the context of axiomatic set theory, the **axiom of the empty set** is as follows:

There exists a set that has no elements:

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

or:

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

The equivalence of these two formulations is proved by Equivalence of Definitions of Empty Set.

## 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.

## Also see

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

## Sources

- 1955: John L. Kelley:
*General Topology*... (previous) ... (next): Chapter $0$: Subsets and Complements; Union and Intersection

- Weisstein, Eric W. "Axiom of the Empty Set." From
*MathWorld*--A Wolfram Web Resource. http://mathworld.wolfram.com/AxiomoftheEmptySet.html