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:


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


Sources