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| In [[axiomatic set theory]] and the branches of [[logic]], [[mathematics]], and [[computer science]] that use it, the '''axiom of pairing''' is one of the [[axiom]]s of [[Zermelo–Fraenkel set theory]].
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| == Formal statement ==
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| In the [[formal language]] of the Zermelo–Fraenkel axioms, the axiom reads:
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| :<math>\forall A \, \forall B \, \exist C \, \forall D \, [ D \in C \iff (D = A \or D = B)]</math>
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| or in words:
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| :[[Given any]] [[Set (mathematics)|set]] ''A'' and any set ''B'', [[Existential quantification|there is]] a set ''C'' such that, given any set ''D'', ''D'' is a member of ''C'' [[if and only if]] ''D'' is [[equal (math)|equal]] to ''A'' [[logical disjunction|or]] ''D'' is equal to ''B''.
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| or in simpler words:
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| :Given two sets, there is a set whose members are exactly the two given sets.
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| == Interpretation ==
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| What the axiom is really saying is that, given two sets ''A'' and ''B'', we can find a set ''C'' whose members are precisely ''A'' and ''B''.
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| We can use the [[axiom of extensionality]] to show that this set ''C'' is unique.
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| We call the set ''C'' the ''pair'' of ''A'' and ''B'', and denote it {''A'',''B''}.
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| Thus the essence of the axiom is:
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| :Any two sets have a pair.
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| {''A'',''A''} is abbreviated {''A''}, called the ''[[singleton (mathematics)|singleton]]'' containing ''A''.
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| Note that a singleton is a special case of a pair.
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| The axiom of pairing also allows for the definition of [[ordered pairs]]. For any sets <math>a</math> and <math>b</math>, the [[ordered pair]] is defined by the following:
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| :<math> (a, b) = \{ \{ a \}, \{ a, b \} \}.\,</math>
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| Note that this definition satisfies the condition
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| :<math>(a, b) = (c, d) \iff a = c \and b = d. </math>
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| Ordered [[tuple|''n''-tuples]] can be defined recursively as follows:
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| :<math> (a_1, \ldots, a_n) = ((a_1, \ldots, a_{n-1}), a_n).\!</math> | |
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| == Non-independence ==
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| The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about any alternative [[axiomatization]] of set theory. Nevertheless, in the standard formulation of the [[Zermelo–Fraenkel set theory]], the axiom of pairing follows from the [[axiom schema of replacement]] applied to any given set with two or more elements, and thus it is sometimes omitted. The existence of such a set with two elements, such as { {}, { {} } }, can be deduced either from the [[axiom of empty set]] and the [[axiom of power set]] or from the [[axiom of infinity]].
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| == Generalisation ==
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| Together with the [[axiom of empty set]], the axiom of pairing can be generalised to the following schema:
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| :<math>\forall A_1 \, \ldots \, \forall A_n \, \exist C \, \forall D \, [D \in C \iff (D = A_1 \or \cdots \or D = A_n)]</math>
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| that is:
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| :Given any [[finite set|finite]] number of sets ''A''<sub>1</sub> through ''A''<sub>''n''</sub>, there is a set ''C'' whose members are precisely ''A''<sub>1</sub> through ''A''<sub>''n''</sub>.
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| This set ''C'' is again unique by the axiom of extension, and is denoted {''A''<sub>1</sub>,...,''A''<sub>''n''</sub>}.
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| Of course, we can't refer to a ''finite'' number of sets rigorously without already having in our hands a (finite) set to which the sets in question belong.
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| Thus, this is not a single statement but instead a [[schema (logic)|schema]], with a separate statement for each [[natural number]] ''n''.
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| *The case ''n'' = 1 is the axiom of pairing with ''A'' = ''A''<sub>1</sub> and ''B'' = ''A''<sub>1</sub>.
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| *The case ''n'' = 2 is the axiom of pairing with ''A'' = ''A''<sub>1</sub> and ''B'' = ''A''<sub>2</sub>.
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| *The cases ''n'' > 2 can be proved using the axiom of pairing and the [[axiom of union]] multiple times.
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| For example, to prove the case ''n'' = 3, use the axiom of pairing three times, to produce the pair {''A''<sub>1</sub>,''A''<sub>2</sub>}, the singleton {''A''<sub>3</sub>}, and then the pair {{''A''<sub>1</sub>,''A''<sub>2</sub>},{''A''<sub>3</sub>}}.
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| The axiom of union then produces the desired result, {''A''<sub>1</sub>,''A''<sub>2</sub>,''A''<sub>3</sub>}. We can extend this schema to include ''n''=0 if we interpret that case as the [[axiom of empty set]].
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| Thus, one may use this as an [[axiom schema]] in the place of the axioms of empty set and pairing. Normally, however, one uses the axioms of empty set and pairing separately, and then proves this as a [[theorem]] schema. Note that adopting this as an axiom schema will not replace the [[axiom of union]], which is still needed for other situations.
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| ==Another alternative==
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| Another axiom which implies the axiom of pairing in the presence of the [[axiom of empty set]] is
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| :<math>\forall A \, \forall B \, \exist C \, \forall D \, [D \in C \iff (D \in A \or D = B)]</math>.
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| Using {} for ''A'' and ''x'' for B, we get {''x''} for C. Then use {''x''} for ''A'' and ''y'' for ''B'', getting {''x,y''} for C. One may continue in this fashion to build up any finite set. And this could be used to generate all [[hereditarily finite set]]s without using the [[axiom of union]].
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| == References ==
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| *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).
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| *Jech, Thomas, 2003. ''Set Theory: The Third Millennium Edition, Revised and Expanded''. Springer. ISBN 3-540-44085-2.
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| *Kunen, Kenneth, 1980. ''Set Theory: An Introduction to Independence Proofs''. Elsevier. ISBN 0-444-86839-9.
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| {{Set theory}}
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| [[Category:Axioms of set theory]]
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| [[de:Zermelo-Fraenkel-Mengenlehre#Die Axiome von ZF und ZFC]]
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