{"id":426,"date":"2007-05-23T14:50:58","date_gmt":"2007-05-23T14:50:58","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/05\/23\/the-axiom-of-pairing\/"},"modified":"2007-05-23T14:50:58","modified_gmt":"2007-05-23T14:50:58","slug":"the-axiom-of-pairing","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/05\/23\/the-axiom-of-pairing\/","title":{"rendered":"The Axiom of Pairing"},"content":{"rendered":"

The axiom of pairing is an interesting beast. It looks simple, and in fact, it
\nis<\/em> simple. But it opens up a range of interesting things that we’d like to be able
\nto do. For example, without the axiom of pairing, we wouldn’t be able to formulate the
\ncartesian products of sets – and without cartesian product, huge ranges of interesting and
\nimportant areas of mathematics would be inaccessible to us. (Note that I’m saying that
\npairing is necessary<\/em>, not that it’s sufficient<\/em>. You also need replacement
\nto get the projection functions that are part of the usual definition of the cartesian
\nproduct.)<\/p>\n

<\/p>\n

So how does the axiom of pairing enable cartesian product?<\/p>\n

The simplest answer to that comes from thinking about what cartesian product
\ndoes<\/em>. Given two sets, A and B, the cartesian product is the set of ordered
\npairs<\/em> where the first element of the pair is an element from A, and the second is an
\nelement from B.<\/p>\n

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

Unfortunately, that definition doesn’t work. A and B can be overlapping sets. Suppose A
\nwas the set {1,2,3}, and B was the set {2,3,4}. Then using the above definition, the pair
\n(2,3), would be represented as the set {2,3}; and the pair (3,2) would be represented as
\nthe set {3,2}. But sets aren’t ordered – so {2,3}={3,2}. But (2,3)≠(3,2). So our naive
\nattempt is no good – it generates unordered<\/em> pairs, where we want ordered
\npairs.<\/p>\n

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

The ordered pair clearly exists: by the axiom of pairing, given a and b, we know that
\nthe set {a,b} exists; and by a second application of pairing, given {a} and {a,b}, we know
\nthat the set {{a},{a,b}} exists. <\/p>\n

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.<\/p>\n

The axiom of pairing does more than just give us a way to do ordered pairs – or even
\nordered tuples. What it does is give us the ability to describe all sorts of
\nstructures<\/em> in terms of sets. It’s sort of the “cons” function of set theory: if we
\ncan find a way to describe a structure in terms in terms of pairings, we can build it with
\nsets. And since we can define ordered lists, unordered collections, pairings, tuples, and
\nmore using pairs as a basis, we can describe pretty much any<\/em> mathematical
\nstructure using set theory – thanks to the axiom of pairing.<\/p>\n","protected":false},"excerpt":{"rendered":"

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 […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[56],"tags":[],"class_list":["post-426","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-6S","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/426","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=426"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/426\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=426"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=426"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=426"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}