# Axiom:Axiom of Empty Set/Set Theory

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## Axiom

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

### Formulation 1

There exists a set that has no elements:

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

### Formulation 2

There exists a set for which membership leads to a contradiction:

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

## 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.