Editing Axiom of extensionality (section)

== In ZF set theory ==
In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:

:<math>\forall x\forall y \, [\forall z \, (\left.z \in x\right. \leftrightarrow \left. z \in y\right.) \rightarrow x=y]</math><ref name=":1">{{Cite web |title=Set Theory > Zermelo-Fraenkel Set Theory (ZF) (Stanford Encyclopedia of Philosophy) |url=https://plato.stanford.edu/entries/set-theory/ZF.html |access-date=2024-11-24 |website=plato.stanford.edu |language=en}}</ref><ref>{{Cite web |title=Zermelo-Fraenkel Set Theory |url=https://www.cs.odu.edu/~toida/nerzic/content/set/ZFC.html |access-date=2024-11-24 |website=www.cs.odu.edu}}</ref><ref>{{Cite web |title=Naive Set Theory |url=https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/sets/sets.html |access-date=2024-11-24 |website=sites.pitt.edu}}</ref>

or in words:
:If the sets <math>x</math> and <math>y</math> have the same members, then they are the same set.<ref name=":1" /><ref name=":0" />

In {{glossary link|pure set theory|glossary=Glossary of set theory}}, all members of sets are themselves sets, but not in set theory with [[urelement]]s. The axiom's usefulness can be seen from the fact that, if one accepts that <math>\exists A \, \forall x \, (x \in A \iff \Phi(x))</math>, where <math>A</math> is a set and ''<math>\Phi(x)</math>'' is a formula that <math>x</math> [[Free variables and bound variables|occurs free]] in but <math>A</math> doesn't, then the axiom assures that there is a unique set <math>A</math> whose members are precisely whatever objects (urelements or sets, as the case may be) satisfy the formula ''<math>\Phi(x)</math>.''

The converse of the axiom, <math>\forall x\forall y \, [x=y \rightarrow \forall z \, (\left.z \in x\right. \leftrightarrow \left. z \in y\right.)]</math>, follows from the [[First-order logic#Equality and its axioms|substitution property]] of [[equality (mathematics)|equality]]. Despite this, the axiom is sometimes given directly as a [[Logical biconditional|biconditional]], i.e., as <math>\forall x\forall y \, [\forall z \, (\left.z \in x\right. \leftrightarrow \left. z \in y\right.) \leftrightarrow x=y]</math>.<ref name=":0" />