Editing Axiom of pairing (section)

== Consequences ==
As noted, what the axiom is saying is that, given two objects ''A'' and ''B'', we can find a set ''C'' whose members are exactly ''A'' and ''B''.

We can use the [[axiom of extensionality]] to show that this set ''C'' is unique. We call the set ''C'' the ''pair'' of ''A'' and ''B'', 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 the ''[[singleton (mathematics)|singleton]]'' containing ''A''. 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 <math>x=\{x\}</math> from the [[Axiom of regularity]].

The axiom of pairing also allows for the definition of [[ordered pairs]]. For any objects <math>a</math> and <math>b</math>, the [[ordered pair]] is defined by the following:

:<math> (a, b) = \{ \{ a \}, \{ a, b \} \}.\,</math>

Note that this definition satisfies the condition

:<math>(a, b) = (c, d) \iff a = c \land b = d. </math>

Ordered [[tuple|''n''-tuples]] can be defined recursively as follows:

:<math> (a_1, \ldots, a_n) = ((a_1, \ldots, a_{n-1}), a_n).\!</math>