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.