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	~~In [[axiomatic set theory]]~~ and ~~the branches of [[logic]]~~, ~~[[mathematics]]~~, and ~~[[computer science]] that use~~ it, ~~the '''axiom of extensionality'''~~, ~~or '''axiom of extension''', is one of the [[axiom]]s of [[Zermelo–Fraenkel set theory]]~~.

	~~== Formal statement ==~~
	~~In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:~~

	:<~~math~~>~~\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow A = B)~~<~~/math~~>

	~~or in words:~~
	~~:[[Given any]] [[Set (mathematics)\|set]] ''A'' and any set ''B'', if for every set ''X'', ''X''~~ is a ~~member~~ of ~~''A'' [[if and only if]] ''X'' is~~ a ~~member~~ of ~~''B'', then ''A'' is [[equal (math)\|equal]]~~ to ~~''B''~~.
	~~:(It~~ is ~~not really essential~~ that ~~''X'' here be a ''set'' — but in '''ZF''', everything is~~. ~~See '''[[#In set theory with ur-elements\|Ur-elements]]''' below for when this~~ is ~~violated~~.)

	~~The converse,~~ <~~math~~>~~\forall A \, \forall B \, (A = B \Rightarrow \forall X \, (X \in A \iff X \in B) )~~<~~/math~~>~~, of this axiom follows from~~ the ~~substitution property of [[equality (mathematics)\|equality]].~~

	~~== Interpretation ==~~
	~~To understand this axiom, note that the clause in parentheses~~ in the ~~symbolic statement above simply states~~ that ~~''A'' and ''B'' have precisely the same members~~.
	~~Thus, what~~ the ~~axiom is really saying is that two sets~~ are ~~equal [[if~~ and ~~only if]] they~~ have ~~precisely the same members.~~
	~~The essence of this~~ is:
	~~:A set is determined uniquely by its members~~.

	~~The axiom of extensionality~~ can be used ~~with~~ any ~~statement of the form~~
	<~~math~~>~~\exist A \, \forall X \, (X \in A \iff P(X) \, )~~<~~/math~~>,
	~~where ''P''~~ is ~~any unary [[Predicate (mathematics)\|predicate]] that does not mention ''A'', to define~~ a ~~unique set <math>A</math> whose members are precisely the sets satisfying the predicate <math>P</math>~~.
	~~We can then introduce a new symbol for <math>~~A~~</math>; it's~~ in ~~this way that [[definition]]s in ordinary mathematics ultimately~~ work ~~when their statements are reduced to purely set-theoretic terms~~.

	~~The axiom of extensionality is generally uncontroversial in set-theoretical foundations of mathematics, and it or an equivalent appears in just about any alternative axiomatisation of set theory.~~
	~~However~~, it ~~may require modifications for some purposes, as below.~~

	~~== In predicate logic without equality ==~~
	~~The axiom given above assumes that equality~~ is ~~a primitive symbol in [[predicate logic]].~~
	~~Some treatments of axiomatic set theory prefer~~ to ~~do without this, and instead treat~~ the ~~above statement not as an axiom but as a ''definition'' of equality.~~
	~~Then~~ it ~~is necessary to include~~ the ~~usual axioms~~ of ~~equality~~ from ~~predicate logic as axioms about this defined symbol. Most~~ of the ~~axioms~~ of ~~equality still follow from the definition; the remaining one~~ is
	~~:<math>\forall A \, \forall B \, ( \forall X \, (X \in A \iff X \in B) \Rightarrow \forall Y \, (A \~~in ~~Y \iff B \in Y) \, )</math>~~
	~~and it becomes ''this'' axiom~~ that ~~is referred to as the axiom of extensionality~~ in ~~this context.~~

	~~== In set theory with ur-elements ==~~
	~~An [[ur-element]] is~~ a ~~member~~ of ~~a set that is not itself a set.~~
	~~In the Zermelo–Fraenkel axioms~~, ~~there are no ur~~-~~elements~~, ~~but they are included in some alternative axiomatisations of set theory~~.
	~~Ur-elements can be treated as a different [[logical type]] from sets; in this case,~~ <~~math~~>~~B \in A~~<~~/math~~> ~~makes no sense if <math>A</math> is an ur-element~~, so the ~~axiom of extensionality simply applies only to sets.~~

	~~Alternatively~~, ~~in untyped logic~~, ~~we can require <math>B \in A</math>~~ to ~~be false whenever <math>A</math> is an ur-element.~~
	In this ~~case, the usual axiom of extensionality would then imply~~ that ~~every ur-element~~ is ~~equal to~~ the ~~[[empty set]].~~
	~~To avoid this consequence, we~~ can ~~modify the axiom of extensionality to apply only to nonempty sets, so that~~ it ~~reads:~~

	~~:<math>\forall A \~~, ~~\forall B \, ( \exist X \, (X \in A) \Rightarrow [ \forall Y \, (Y \in A \iff Y \in B) \Rightarrow A = B ] \, ).</math>~~

	~~That is:~~
	~~:Given any set ''A''~~ and any ~~set ''B'', ''if ''A'' is a nonempty set'' (~~that ~~is, if there exists a member ''X'' of ''A''), ''then'' if ''A''~~ and '~~'B'' have precisely~~ the ~~same members, then they are equal~~.

	~~Yet another alternative in untyped logic is to define <math>~~A~~</math> itself~~ to ~~be the only element of~~ <~~math~~>A<~~/math~~>
	~~whenever <math>A</math> is an ur-element. While~~ this ~~approach can serve~~ to ~~preserve the axiom of extensionality, the [[axiom of regularity]] will need an adjustment instead.~~

	~~== See also ==~~
	* [[Extensionality]] for a general overview.

	~~== References ==~~
	*[[~~Paul Halmos]], ''Naive set theory''. Princeton, NJ~~: D. ~~Van Nostrand Company, 1960~~. ~~Reprinted by Springer-Verlag, New York, 1974. ISBN 0-387-90092-6 (Springer-Verlag edition).~~
	*[[Thomas Jech\|Jech, Thomas]]~~, 2003~~. ~~''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.~~
	*[[Kenneth Kunen\|Kunen, Kenneth]], 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.

	~~{{Set theory}}~~

	~~[[Category:Axioms of set theory]]~~
	~~[[Category:Urelements]]~~

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