# Axiom:Axiom of 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

### 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:

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

### 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.