Editing Axiom of extensionality

{{short description|Axiom used in set theory}}
{{About|the axiom used in formal set theory|the article on ''extensionality'' in general|Extensionality}}

The '''axiom of extensionality''',<ref name=":0">{{Cite web |title=AxiomaticSetTheory |url=https://www.cs.yale.edu/homes/aspnes/pinewiki/AxiomaticSetTheory.html |access-date=2024-08-20 |website=www.cs.yale.edu}}</ref><ref>{{Cite web |title=Naive Set Theory |url=https://sites.pitt.edu/~jdnorton/teaching/paradox/chapters/sets/sets.html |access-date=2024-08-20 |website=sites.pitt.edu}}</ref> also called the '''axiom of extent''',<ref>{{Cite book |last=Bourbaki |first=N. |url=https://books.google.com/books?id=7eclBQAAQBAJ |title=Theory of Sets |date=2013-12-01 |publisher=Springer Science & Business Media |isbn=978-3-642-59309-3 |pages=67 |language=en}}</ref><ref>{{Cite book |last=Deskins |first=W. E. |url=https://books.google.com/books?id=2KrDAgAAQBAJ |title=Abstract Algebra |date=2012-05-24 |publisher=Courier Corporation |isbn=978-0-486-15846-4 |pages=2 |language=en}}</ref> is an [[axiom]] used in many forms of [[axiomatic set theory]], such as [[Zermelo–Fraenkel set theory]].<ref>{{Cite web |title=Zermelo-Fraenkel Set Theory |url=https://www.cs.odu.edu/~toida/nerzic/content/set/ZFC.html |access-date=2024-08-20 |website=www.cs.odu.edu}}</ref><ref>{{Cite web |title=Intro to Axiomatic (ZF) Set Theory |url=https://www.andrew.cmu.edu/user/kk3n/complearn/sets.html |access-date=2024-08-20 |website=www.andrew.cmu.edu}}</ref> The axiom defines what a [[Set (mathematics)|set]] is.<ref name=":0" /> Informally, the axiom means that the two [[set (mathematics)|set]]s ''A'' and ''B'' are equal [[if and only if]] ''A'' and ''B'' have the same members.

== Etymology ==
The term ''[[extensionality]]'', as used in ''<nowiki/>'Axiom of Extensionality''' has its roots in logic. An [[intensional definition]] describes the [[necessary and sufficient]] conditions for a term to apply to an object. For example: "An [[even number]] is an [[integer]] which is [[divisible]] by 2." An extensional definition instead lists all objects where the term applies. For example: "An even number is any one of the following integers: 0, 2, 4, 6, 8..., -2, -4, -6, -8..." In logic, the [[Extension (logic)|extension]] of a [[Predicate (mathematical logic)|predicate]] is the set of all things for which the predicate is true.<ref>{{Cite book |last=Roy T Cook |url=https://archive.org/details/roy-t.-cook-a-dictionary-of-philosophical-logic/page/155/mode/2up?q=%22INTENSIONAL+DEFINITION%22 |title=A Dictionary Of Philosophical Logic |date=2010 |isbn=978-0-7486-2559-8 |pages=155}}</ref>

The logical term was introduced to set theory in 1893, [[Gottlob Frege]] attempted to use this idea of an extension formally in his [[Basic Laws of Arithmetic|''Basic Laws of Arithmetic'']] (German: ''Grundgesetze der Arithmetik''),<ref>{{Cite book |last=Lévy |first=Azriel |url=https://archive.org/details/basicsettheory00levy_0/mode/2up?q=Frege |title=Basic set theory |date=1979 |publisher=Berlin ; New York : Springer-Verlag |isbn=978-0-387-08417-6 |pages=5}}</ref><ref>{{Cite book |last=Frege |first=Gottlob |url=https://archive.org/details/bub_gb_LZ5tAAAAMAAJ/page/n105/ |title=Grundgesetze der arithmetik |date=1893 |publisher=Jena, H. Pohle |pages=69}}</ref> where, if <math>F</math> is a predicate, its extension (German: ''Umfang'') <math>\varepsilon F</math>, is the set of all objects satisfying <math>F</math>.<ref>{{Citation |last=Zalta |first=Edward N. |title=Frege's Theorem and Foundations for Arithmetic |date=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/frege-theorem/ |access-date=2025-01-16 |edition=Spring 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref> For example if <math>F(x)</math> is "x is even" then <math>\varepsilon F</math> is the set <math>\{ \cdots , -4, -2, 0, 2, 4, \cdots \} </math>. In his work, he defined his infamous ''[[Basic Law V]]'' as:{{Sfn|Ferreirós|2007|p=304}}<math display="block">\varepsilon F = \varepsilon G \equiv \forall x (F(x) \equiv G(x) ) </math>Stating that if two predicates have the same extensions (they are satisfied by the same set of objects) then they are logically equivalent, however, it was determined later that this axiom led to [[Russell's paradox]]. The first explicit statement of the modern Axiom of Extensionality was in 1908 by Ernst Zermelo in a paper on the [[well-ordering theorem]], where he presented the first axiomatic set theory, now called [[Zermelo set theory]], which became the basis of modern set theories.<ref>{{Citation |last=Hallett |first=Michael |title=Zermelo's Axiomatization of Set Theory |date=2024 |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/zermelo-set-theory/ |access-date=2025-01-16 |edition=Fall 2024 |publisher=Metaphysics Research Lab, Stanford University |editor2-last=Nodelman |editor2-first=Uri |encyclopedia=The Stanford Encyclopedia of Philosophy}}</ref> The specific term for "Extensionality" used by Zermelo was "Bestimmtheit".The specific English term "extensionality" only became common in mathematical and logical texts in the 1920s and 1930s,<ref>[[Oxford English Dictionary]], s.v. “[[doi:10.1093/OED/1191853349|Extensionality (n.)]]” December 2024</ref> particularly with the formalization of logic and set theory by figures like [[Alfred Tarski]] and [[John von Neumann]].

== 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" />

== In NF set theory ==
[[Willard Van Orman Quine|Quine]]'s [[New Foundations]] (NF) set theory, in Quine's original presentations of it, treats the symbol <math>=</math> for equality or identity as shorthand either for "if a set contains the left side of the equals sign as a member, then it also contains the right side of the equals sign as a member" (as defined in 1937), or for "an object is an element of the set on the left side of the equals sign if, and only if, it is also an element of the set on the right side of the equals sign" (as defined in 1951). That is, <math>x=y</math> is treated as shorthand either for <math>\forall z \, \left.(x \in z\right. \rightarrow \left. y\in z\right.)</math>, as in the original 1937 paper, or for <math>\forall z \, \left.(z \in x\right. \leftrightarrow \left. z\in y\right.)</math>, as in Quine's ''Mathematical Logic'' (1951). The second version of the definition is exactly equivalent to the [[Antecedent (logic)|antecedent]] of the ZF axiom of extensionality, and the first version of the definition is still very similar to it. By contrast, however, the ZF set theory takes the symbol <math>=</math> for identity or equality as a primitive symbol of the formal language, and defines the axiom of extensionality in terms of it. (In this paragraph, the statements of both versions of the definition were paraphrases, and quotation marks were only used to set the statements apart.)

In Quine's ''New Foundations for Mathematical Logic'' (1937), the original paper of NF, the name "''principle of extensionality''" is given to the postulate P1, <math>( ( x \subset y ) \supset ( ( y \subset x ) \supset (x = y) ) )</math>,<ref name=":2">{{Cite journal |last=Quine |first=W. V. |date=1937 |title=New Foundations for Mathematical Logic |url=https://www.jstor.org/stable/2300564 |journal=The American Mathematical Monthly |volume=44 |issue=2 |pages=74, 77 |doi=10.2307/2300564 |jstor=2300564 |issn=0002-9890|url-access=subscription }}</ref> which, for readability, may be restated as <math> x \subset y \rightarrow ( y \subset x \rightarrow x = y )</math>. The definition D8, which defines the symbol <math>=</math> for identity or equality, defines <math>(\alpha = \beta)</math> as shorthand for <math>(\gamma) \, (\left.(\alpha \in \gamma\right.) \supset (\left. \beta\in \gamma\right.))</math>.<ref name=":2" /> In his ''Mathematical Logic'' (1951), having already developed [[quasi-quotation]], Quine defines <math>\ulcorner \zeta=\eta \urcorner</math> as shorthand for <math>\ulcorner (\alpha) \, (\left.\alpha \in \zeta\right. \; . \equiv  \, .  \, \left. \alpha \in \eta\right.) \urcorner</math> (definition D10), and does not define an axiom or principle "of extensionality" at all.<ref>{{Cite journal |last=Quine |first=W. V. |date=1951-12-31 |title=Mathematical Logic |url=http://dx.doi.org/10.4159/9780674042469 |journal=DeGruyter |pages=134–136 |doi=10.4159/9780674042469|isbn=978-0-674-04246-9 |url-access=subscription }}</ref>

[[Thomas Forster (mathematician)|Thomas Forster]], however, ignores these fine distinctions, and considers NF to accept the axiom of extensionality in its ZF form.<ref>{{Citation |last=Forster |first=Thomas |title=Quine's New Foundations |date=2019 |encyclopedia=The Stanford Encyclopedia of Philosophy |editor-last=Zalta |editor-first=Edward N. |url=https://plato.stanford.edu/entries/quine-nf/ |access-date=2024-11-24 |edition=Summer 2019 |publisher=Metaphysics Research Lab, Stanford University}}</ref>

== In ZU set theory ==
In the [[Scott–Potter set theory|Scott–Potter]] (ZU) set theory, the "extensionality principle" <math>( \forall x ) (\left.x \in a\right. \Leftrightarrow \left. x \in b\right.) \Rightarrow a=b</math> is given as a theorem rather than an axiom, which is proved from the definition of a "collection".<ref>{{Cite book |last=Potter |first=Michael D. |url=https://www.worldcat.org/title/ocm53392572 |title=Set theory and its philosophy: a critical introduction |date=2004 |publisher=Oxford University Press |isbn=978-0-19-926973-0 |location=Oxford; New York |pages=31 |oclc=ocm53392572}}</ref>

== In set theory with ur-elements ==
{{Unreferenced section|date=November 2024}}
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 \, ( \exists X \, (X \in A) \implies [ \forall Y \, (Y \in A \iff Y \in B) \implies 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]]
* [[Extensional context]]
* [[Extension (predicate logic)]]
* [[Set theory]]
* [[Glossary of set theory]]

== References ==
{{refbegin}}
* {{Citation |last=Ferreirós |first=José |title=Labyrinth of Thought: A History of Set Theory and Its Role in Mathematical Thought |year=2007 |edition=2nd revised |publisher=[[Birkhäuser]] |isbn=978-3-7643-8349-7}}
*[[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}}.
{{refend}}

===Notes===
{{reflist}}

{{Set theory}}

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