Editing Equality (mathematics) (section)

== In set theory ==
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[[Set theory]] is the branch of mathematics that studies [[Set (mathematics)|sets]], which can be informally described as "collections of objects".<ref>{{Cite book |last=Breuer |first=Josef |url=https://archive.org/details/introductiontoth00breu/mode/2up |title=Introduction to the Theory of Sets |date=1958 |publisher=Prentice-Hall |location=Englewood Cliffs, New Jersey |via=Internet Archive |page=4 |quote=A set is a collection of definite distinct objects of our perception or of our thought, which are called elements of the set.}}</ref> Although objects of any kind can be collected into a set, set theory{{snd}}as a branch of mathematics{{snd}}is mostly concerned with those that are relevant to mathematics as a whole.
Sets are uniquely characterized by their [[Element (mathematics)|elements]]; this means that two sets that have precisely the same elements are equal (they are the same set).{{Sfn|Stoll|1963|pp=4–5}} In a [[Formal set theory|formalized set theory]], this is usually defined by an [[axiom]] called the [[Axiom of extensionality]].<ref>{{harvnb|Lévy|2002|pp=13, 358}}. {{harvnb|Mac Lane|Birkhoff|1999|p=2}}. {{harvnb|Mendelson|1964|p=5}}.</ref>

For example, using [[set builder notation]], the following states that "The set of all [[Integer|integers]] <math>(\Z)</math> greater than 0 but not more than 3 is equal to the set containing only 1, 2, and 3", despite the differences in notation. 

<math display="block">\{x\in \Z \mid 0<x\le 3\} = \{1,2,3\},</math>

The term ''extensionality'', as used in ''<nowiki/>'Axiom of Extensionality''' has its roots in logic and grammar (''cf. [[Extension (semantics)]]''). In grammar, [[intensional definition]] describes the [[necessary and sufficient]] conditions for a term to apply to an object. For example: "A [[Platonic solid]] is a [[Convex polytope|convex]], [[regular polyhedron]] in [[Three-dimensional space|three-dimensional Euclidean space]]." An extensional definition instead lists all objects where the term applies. For example: "A Platonic solid is one of the following: [[Tetrahedron]], [[Cube]], [[Octahedron]], [[Dodecahedron]], or [[Icosahedron]]." In logic, the [[Extension (logic)|extension]] of a [[Predicate (mathematical logic)|predicate]] is the set of all objects for which the predicate is true.<ref>{{Cite book |last=Cook |first=Roy T. |url=https://edinburghuniversitypress.com/book-a-dictionary-of-philosophical-logic.html |title=A Dictionary Of Philosophical Logic |date=2009 |publisher=[[Edinburgh University Press]] |isbn=978-0-7486-2559-8 |location=Edinburgh |pages=155 |archive-url=https://archive.org/details/roy-t.-cook-a-dictionary-of-philosophical-logic/ |archive-date=2021-05-14}}</ref> Further, the logical principle of [[extensionality]] judges two objects to objects to be equal if they satisfy the same external properties. Since, by the axiom, two sets are defined to be equal if they satisfy [[Membership (set theory)|membership]], sets are extentional.<ref>{{Cite book |last=Mayberry |first=John P. |url=https://archive.org/details/foundationsofmat0000john |title=Foundations of Mathematics in the Theory of Sets |date=2011 |publisher=[[Cambridge University Press]] |isbn=978-0-521-17271-4 |series=Encyclopedia of Mathematics and its Applications |location=New York |pages=74, 113 |doi=10.1017/CBO9781139087124}}</ref>

[[José Ferreirós]] credits [[Richard Dedekind]] for being the first to explicitly state the principle, although he does not assert it as a definition:{{Sfn|Ferreirós|2007|p=226}}
{{blockquote|It very frequently happens that different things a, b, c... considered for any reason under a common point of view, are collected together in the mind, and one then says that they form a system S; one calls the things a, b, c... the elements of the system S, they are contained in S; conversely, S consists of these elements. Such a system S (or a collection, a manifold, a totality), as an object of our thought, is likewise a thing; it is completely determined when, for every thing, it is determined whether it is an element of S or not.
|source=Richard Dedekind, 1888 (translated by José Ferreirós)
}}

=== Background ===
{{further|Set theory#History}}
[[File:Ernst Zermelo 1900s.jpg|thumb|[[Ernst Zermelo]], a contributor to modern Set theory, was the first to explicitly formalize set equality in his [[Zermelo set theory]] (now obsolete), by his ''{{lang|de|Axiom der Bestimmtheit}}''.<ref>{{citation |last=Zermelo |first=Ernst |title=Untersuchungen über die Grundlagen der Mengenlehre I |journal=Mathematische Annalen |volume=65 |issue=2 |pages=261–281 |year=1908 |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=PPN235181684_0065&DMDID=DMDLOG_0018 |doi=10.1007/bf01449999 |s2cid=120085563 |author-link=Ernst Zermelo |language=de}}</ref>]]
Around the turn of the 20th century, mathematics faced several [[paradox]]es and counter-intuitive results. For example, [[Russell's paradox]] showed a contradiction of [[naive set theory]], it was shown that the [[parallel postulate]] cannot be proved, the existence of [[mathematical object]]s that cannot be computed or explicitly described, and the existence of theorems of arithmetic that cannot be proved with [[Peano arithmetic]]. The result was a [[foundational crisis of mathematics]].{{Sfn|Ferreirós|2007|p=299}}

The resolution of this crisis involved the rise of a new mathematical discipline called [[mathematical logic]], which studies [[Logic#Formal logic|formal logic]] within mathematics. Subsequent discoveries in the 20th century then stabilized the foundations of mathematics into a coherent framework valid for all mathematics. This framework is based on a systematic use of [[axiomatic method]] and on set theory, specifically [[Zermelo–Fraenkel set theory]], developed by [[Ernst Zermelo]] and [[Abraham Fraenkel]]. This set theory (and set theory in general) is now considered the most common [[foundation of mathematics]].{{Sfn|Ferreirós|2007|p=366|loc="[...] the most common axiom system was and is called the Zermelo-Fraenkel system."}}

=== Set equality based on first-order logic with equality ===
In first-order logic with equality (See {{Section link||Axioms}}), the axiom of extensionality states that two sets that ''contain'' the same elements are the same set.<ref>{{harvnb|Kleene|1967|page=189}}. {{harvnb|Lévy|2002|page=13}}. {{harvnb|Shoenfield|2001|page=239}}.</ref>
* Logic axiom: <math>x = y \implies \forall z, (z \in x \iff z \in y)</math>
* Logic axiom: <math>x = y \implies \forall z, (x \in z \iff y \in z)</math>
* Set theory axiom: <math>(\forall z, (z \in x \iff z \in y)) \implies x = y</math>

The first two are given by the substitution property of equality from first-order logic; the last is a new axiom of the theory. Incorporating half of the work into the first-order logic may be regarded as a mere matter of convenience, as noted by [[Azriel Lévy]].
: "The reason why we take up first-order predicate calculus ''with equality'' is a matter of convenience; by this, we save the labor of defining equality and proving all its properties; this burden is now assumed by the logic."<ref>{{harvnb|Lévy|2002|page=4}}.</ref>

=== Set equality based on first-order logic without equality ===
In first-order logic without equality, two sets are ''defined'' to be equal if they contain the same elements. Then the axiom of extensionality states that two equal sets ''are contained in'' the same sets.<ref>{{harvnb|Mendelson|1964|pages=159–161}}.{{harvnb|Rosser|2008|pages=211–213}}</ref>
* Set theory definition:  <math>(x = y) \ := \ \forall z, (z \in x \iff z \in y)</math>
* Set theory axiom: <math>x = y \implies \forall z, (x \in z \iff y \in z)</math>
Or, equivalently, one may choose to define equality in a way that mimics, the substitution property explicitly, as the [[Logical conjunction|conjunction]] of all [[Atomic formula|atomic formuals]]:<ref>{{Cite book |last=Fraenkel |first=Abraham Adolf |url=https://archive.org/details/foundationsofset0000frae/page/26/mode/2up?q=Extensionality |title=Foundations of set theory |date=1973 |publisher=Noord-Holland |location=Amsterdam |isbn=978-0-7204-2270-2 |volume=67 |edition=2nd revised |page=27 |oclc=731740381}}</ref>
* Set theory definition: <math>(x = y) := </math><math> \; \forall z ( z \in x \implies z \in y) \, \and \, </math><math> \forall w (x \in w \implies y \in w)</math>
* Set theory axiom: <math>(\forall z, (z \in x \iff z \in y)) \implies x = y</math>

In either case, the Axiom of Extensionality based on first-order logic without equality states:

<math display="block">\forall z ( z \in x \Rightarrow z \in y) \implies \forall w (x \in w \Rightarrow y \in w) </math>

=== Proof of basic properties ===
* '''Reflexivity:''' Given a set <math>X,</math> assume <math>z \in X,</math> it follows trivially that <math>z \in X,</math> and the same follows in reverse, therefore <math>\forall z, (z \in X \iff z \in X)</math> , thus <math>X = X.</math><ref name="Takeuti1982">{{Cite book |last1=Takeuti |first1=Gaisi |last2=Zaring |first2=Wilson M. |date=1982 |title=Introduction to Axiomatic Set Theory |publisher=Springer |location=New York |series=Graduate Texts in Mathematics |volume=1 |page=7 |doi=10.1007/978-1-4613-8168-6 |isbn=978-1-4613-8170-9 |issn=0072-5285}}</ref>
* '''Symmetry:''' Given sets <math>X, Y,</math> such that <math>X=Y,</math> then <math>\forall z, (z \in X \iff z \in Y),</math> which implies <math>\forall z, (z \in Y \iff z \in X),</math> thus <math>Y=X.</math><ref name="Takeuti1982" />
* '''Transitivity:''' Given sets <math>X, Y, Z,</math> such that (1) <math>X = Y</math> and (2) <math>Y = Z,</math> assume <math>z \in X,</math> then <math>z \in Y</math> by (1), which implies <math>z \in Z</math> by (2), and similarly for the reverse, therefore <math>\forall z, (z \in X \iff z \in Z)  ,</math> thus <math>X=Z.</math><ref name="Takeuti1982" />
* '''Substitution:''' ''See {{Section link|Substitution (logic)|Proof of substitution in ZFC}}.''
* '''Function application:''' Given <math>a = b</math> and <math>f(a) = c,</math> then <math>(a,c) \in f.</math> Since <math>a = b</math> and <math>c = c,</math> then <math>(a,c) = (b,c).</math> This is the defining property of an [[ordered pair]].{{Snf|Stoll|1963|p=24}} Since <math>(a,c) = (b,c),</math> by the Axiom of Extensionality, they must belong to the same sets, so, since <math>(a,c) \in f,</math> we have <math>(b,c) \in f,</math> or <math>f(b) = c.</math> Thus <math>f(a) = f(b).</math>