The Axiom of Pairing

The axiom of pairing is an interesting beast. It looks simple, and in fact, it
is simple. But it opens up a range of interesting things that we'd like to be able
to do. For example, without the axiom of pairing, we wouldn't be able to formulate the
cartesian products of sets - and without cartesian product, huge ranges of interesting and
important areas of mathematics would be inaccessible to us. (Note that I'm saying that
pairing is necessary, not that it's sufficient. You also need replacement
to get the projection functions that are part of the usual definition of the cartesian
product.)

So how does the axiom of pairing enable cartesian product?

The simplest answer to that comes from thinking about what cartesian product
does. Given two sets, A and B, the cartesian product is the set of ordered
pairs where the first element of the pair is an element from A, and the second is an
element from B.

What's an ordered pair in terms of sets? Naively, you might come up with something like
if a∈A, and b∈B, then the ordered pair (a,b) would be the set containing a and b:
{a,b}. And clearly, the axiom of pairing does guarantee that we can do that: it
says that if a and b are sets, then {a,b} is a set.

Unfortunately, that definition doesn't work. A and B can be overlapping sets. Suppose A
was the set {1,2,3}, and B was the set {2,3,4}. Then using the above definition, the pair
(2,3), would be represented as the set {2,3}; and the pair (3,2) would be represented as
the set {3,2}. But sets aren't ordered - so {2,3}={3,2}. But (2,3)≠(3,2). So our naive
attempt is no good - it generates unordered pairs, where we want ordered
pairs.

So our representation of ordered pairs needs to have some way of distinguishing which
element of a pair came first. How do we say which element of a pair comes first? Well, if
we've got an unordered pair {a,b}, the way we can say which element came first is
by created another unordered pair, which contains two sets: the first
unordered pair, and a set containing the member of that unordered pair which
should come first. That sounds a bit confusing, but it's clear once you see an example. If
(a,b) is an ordered pair, then {a,b} is the unordered pair containing a and b. To
make it ordered, we create the pair {{a},{a,b}}. {{a},{a,b}} is the set representation of
the ordered pair. We can tell which member of the set representation identifies the first
element of the ordered pair by using a subset test: if {x,y} is the set representation of
an ordered pair, then either x⊂y or y⊂x. If x⊂y, then x identifies the first
element of the pair; otherwise, y does.

The ordered pair clearly exists: by the axiom of pairing, given a and b, we know that
the set {a,b} exists; and by a second application of pairing, given {a} and {a,b}, we know
that the set {{a},{a,b}} exists.

It's worth pointing out here that since set theory considers a function to be nothing more than a collection of ordered pairs that construction above also means that the axiom of pairing allows us to define functions in terms of sets.

The axiom of pairing does more than just give us a way to do ordered pairs - or even
ordered tuples. What it does is give us the ability to describe all sorts of
structures in terms of sets. It's sort of the "cons" function of set theory: if we
can find a way to describe a structure in terms in terms of pairings, we can build it with
sets. And since we can define ordered lists, unordered collections, pairings, tuples, and
more using pairs as a basis, we can describe pretty much any mathematical
structure using set theory - thanks to the axiom of pairing.