# Category:Axioms/Axiom of Empty Set

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This category contains axioms related to Axiom of Empty Set.

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.

## Pages in category "Axioms/Axiom of Empty Set"

The following 6 pages are in this category, out of 6 total.