Axiom of pairing

Inaxiomatic set theoryand the branches oflogic,mathematics,andcomputer sciencethat use it, theaxiom of pairingis one of theaxiomsofZermelo–Fraenkel set theory.It was introduced byZermelo (1908)as a special case of hisaxiom of elementary sets.

Formal statement

In theformal languageof the Zermelo–Fraenkel axioms, the axiom reads:

$\forall A\,\forall B\,\exists C\,\forall D\,[D\in C\iff (D=A\lor D=B)]$

In words:

Given anyobjectAand any objectB,there isa setCsuch that, given any objectD,Dis a member ofCif and only ifDisequaltoAorDis equal toB.

Consequences

As noted, what the axiom is saying is that, given two objectsAandB,we can find a setCwhose members are exactlyAandB.

We can use theaxiom of extensionalityto show that this setCis unique. We call the setCthepairofAandB,and denote it {A,B}. Thus the essence of the axiom is:

Any two objects have a pair.

The set {A,A} is abbreviated {A}, called thesingletoncontainingA. Note that a singleton is a special case of a pair. Being able to construct a singleton is necessary, for example, to show the non-existence of the infinitely descending chains$x=\{x\}$from theAxiom of regularity.

The axiom of pairing also allows for the definition ofordered pairs.For any objects$a$and$b$,theordered pairis defined by the following:

$(a,b)=\{\{a\},\{a,b\}\}.\,$

Note that this definition satisfies the condition

$(a,b)=(c,d)\iff a=c\land b=d.$

Orderedn-tuplescan be defined recursively as follows:

$(a_{1},\ldots,a_{n})=((a_{1},\ldots,a_{n-1}),a_{n}).\!$

Alternatives

Non-independence

The axiom of pairing is generally considered uncontroversial, and it or an equivalent appears in just about anyaxiomatizationof set theory. Nevertheless, in the standard formulation of theZermelo–Fraenkel set theory,the axiom of pairing follows from theaxiom schema of replacementapplied 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 theaxiom of empty setand theaxiom of power setor from theaxiom of infinity.

In the absence of some of the stronger ZFC axioms, the axiom of pairing can still, without loss, be introduced in weaker forms.

Weaker

In the presence of standard forms of theaxiom schema of separationwe can replace the axiom of pairing by its weaker version:

$\forall A\forall B\exists C\forall D((D=A\lor D=B)\Rightarrow D\in C)$.

This weak axiom of pairing implies that any given objects$A$and$B$are members of some set$C$.Using the axiom schema of separation we can construct the set whose members are exactly$A$and$B$.

Another axiom which implies the axiom of pairing in the presence of theaxiom of empty setis theaxiom of adjunction

$\forall A\,\forall B\,\exists C\,\forall D\,[D\in C\iff (D\in A\lor D=B)]$.

It differs from the standard one by use of$D\in A$instead of$D=A$. Using {} forAandxfor B, we get {x} for C. Then use {x} forAandyforB,getting {x,y} for C. One may continue in this fashion to build up any finite set. And this could be used to generate allhereditarily finite setswithout using theaxiom of union.

Stronger

Together with theaxiom of empty setand theaxiom of union,the axiom of pairing can be generalised to the following schema:

$\forall A_{1}\,\ldots \,\forall A_{n}\,\exists C\,\forall D\,[D\in C\iff (D=A_{1}\lor \cdots \lor D=A_{n})]$

that is:

Given anyfinitenumber of objectsA₁throughA_n,there is a setCwhose members are preciselyA₁throughA_n.

This setCis again unique by theaxiom of extensionality,and is denoted {A₁,...,A_n}.

Of course, we can't refer to afinitenumber of objects rigorously without already having in our hands a (finite) set to which the objects in question belong. Thus, this is not a single statement but instead aschema,with a separate statement for eachnatural numbern.

• The casen= 1 is the axiom of pairing withA=A₁andB=A₁.
• The casen= 2 is the axiom of pairing withA=A₁andB=A₂.
• The casesn> 2 can be proved using the axiom of pairing and theaxiom of unionmultiple times.

For example, to prove the casen= 3, use the axiom of pairing three times, to produce the pair {A₁,A₂}, the singleton {A₃}, and then the pair {{A₁,A₂},{A₃}}. Theaxiom of unionthen produces the desired result, {A₁,A₂,A₃}. We can extend this schema to includen=0 if we interpret that case as theaxiom of empty set.

Thus, one may use this as anaxiom schemain 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 atheoremschema. Note that adopting this as an axiom schema will not replace theaxiom of union,which is still needed for other situations.

References

• Paul Halmos,Naive set theory.Princeton, NJ: D. Van Nostrand Company, 1960. Reprinted by Springer-Verlag, New York, 1974.ISBN0-387-90092-6(Springer-Verlag edition).
• Jech, Thomas, 2003.Set Theory: The Third Millennium Edition, Revised and Expanded.Springer.ISBN3-540-44085-2.
• Kunen, Kenneth, 1980.Set Theory: An Introduction to Independence Proofs.Elsevier.ISBN0-444-86839-9.
• Zermelo, Ernst(1908),"Untersuchungen über die Grundlagen der Mengenlehre I",Mathematische Annalen,65(2): 261–281,doi:10.1007/bf01449999,S2CID120085563.English translation:Heijenoort, Jean van(1967), "Investigations in the foundations of set theory",From Frege to Gödel: A Source Book in Mathematical Logic, 1879-1931,Source Books in the History of the Sciences, Harvard Univ. Press, pp. 199–215,ISBN978-0-674-32449-7.