{"id":424,"date":"2007-05-20T22:08:50","date_gmt":"2007-05-20T22:08:50","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/05\/20\/the-axioms-of-set-theory\/"},"modified":"2007-05-20T22:08:50","modified_gmt":"2007-05-20T22:08:50","slug":"the-axioms-of-set-theory","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/05\/20\/the-axioms-of-set-theory\/","title":{"rendered":"The Axioms of Set Theory"},"content":{"rendered":"

Axiomatic set theory builds up set theory from a set of fundamental initial rules. The most common axiomatization, which we’ll be used, is the ZFC system: Zermelo-Fraenkel<\/em> with choice<\/em> set theory. The ZFC axiomatization consists of 8 basic rules which are pretty much universally accepted, and two rules that are somewhat controversial – most particularly the last rule, called the axiom of choice<\/em>.<\/p>\n

<\/p>\n

There’s an interesting parallel between the axioms of ZFC and the axioms of Euclidean logic. Euclidean logic consists of a set of universally accepted rules, and one strange one – the parallel axiom – which was problematic for many years, until someone finally realized that you could create a variety of alternate geometries by using variations on the parallel axiom. Similarly, with set theory, there’s the initial set of basic axioms, and then there’s the axiom of choice; if you throw away the axiom of choice, you can construct set theories with different properties.<\/p>\n

Now, it’s time for a first look at the basic list of axioms. An important thing to remember as you look at them is that axiomatic set theory is intended to be a foundational theory of mathematics – and so the only objects that exist in the domain of the theory are sets – no numbers, no functions, nothing. It’s sets, all the way down.<\/p>\n

\n
The Axiom of Extensionality<\/b>: ∀A,B: A=B ⇒ (∀C: C∈A ⇒ C∈B). <\/dt>\n
This is a formal way of saying that a set is described by its members: two sets are equivalent if and only if they contain the same members.<\/dd>\n
The Empty-Set Axiom<\/b>: ∃∅: ∀X: X∉∅.<\/dt>\n
There exists a set, the empty set, which contains no members.<\/dd>\n
The Axiom of Pairing<\/b>: ∀A,B: (∃C: (∀D: D∈C ⇒ (D=A ∨ D=B)))<\/dt>\n
This one is very hairy in logical form. What it really says is that given any two sets A and B, there’s a set C containing only<\/em> A and B as members. Grind through the logical form, and that’s what it says: for any sets A and B, there’s a set C, and any member of C is equal to either A or B.<\/dd>\n
The Axiom of Union<\/b>: ∀A: ∃B: ∀C: C∈B ⇒ (∃D: C∈D ∧ D∈A)<\/dt>\n
Once again, the precise formal statement in FOPL is tough going. There’s an easier way to write it using the union symbol: ∀A: ∃B : B=∪<\/b>a∈A<\/sub>a; that is, for any set A, there’s a set B consisting of the unions the members of A.<\/dd>\n
The Axiom of Infinity<\/b>: &exists;N: ∅∈N ∧ (∀x: x∈N ⇒ x∪{x}isin;N)<\/dt>\n
This is the first of the axioms that I think is actually difficult. Some of the earlier ones can be hard to read, but once you get through the notation, they’re really pretty simple. This one, the notation isn’t so bad, but the meaning is a bugger. What is says is, there’s a set that (a) contains the empty set as a member, and (b) for each of its members x, it also<\/em> contains the singleton set {x} containing x. So, if following the formal statement, we called that set N, N contains ∅, {∅},{{∅}}, {{{∅}}}, etc. What the axiom of infinity does is really two basic things: it gives us our first countably infinite set; and it gives us a construction which can be turned into Peano integers.<\/dd>\n
The Meta-Axiom of Specification<\/b>: ∀A: &exists;B: ∀C: C∈B ⇒ C∈A ∧ P(C)<\/dt>\n
This is pretty simple, but it’s got one trick. The axiom of specification is really a second-order axiom. But in order to make it work in a first-order logical framework, we cheat, and say it’s a scheme<\/em> for what’s actually an infinite set of axioms – for any predicate P, there’s another instantiation of the axiom of specification. What it says is that given a set A and a predicate P, there’s a set B consisting of the members of A for which P is true. Another way of saying that is that for any set A, we can define a subset of A using a predicate which selects members of A.<\/dd>\n
The Meta-Axiom of Replacement<\/b>: ∀A: ∃B: ∀y: y∈B ⇒ ∃x∈A: y=F(x).<\/dt>\n
This is another meta-axiom. It’s a tricky one in its way. One way of looking at it as saying that you can define functions in terms of predicates: P(x,y) is a function if the set defined by it has the necessary properties. We’ll talk more about it later.<\/dd>\n
The Powerset Construction Axiom<\/b>: ∀A: ∃B: ∀C⊆A: C∈B<\/dt>\n
This is a nice, easy one. For any set A, the powerset – that is, the class of all subsets of A – is a set.<\/dd>\n
The Foundation Axiom:<\/b> ∀A≠∅: ∃B∈A: A∩B=∅<\/dt>\n
This one has had its moments of controversy, but it’s really pretty simple in concept. Every set A contains some member B which is a set completely disjoint from A.<\/dd>\n
The Axiom of Choice<\/b>: ∀X: ((∀A∈X: A≠∅) ∧
\n(∀B,C≠∈X: B∩C=∅)) ⇒
\n(∃Y: ∀I∈X:∃!J∈Y: J∈I)<\/dt>\n
For any disjoint<\/em> set of non-empty sets X, there is a set Y called the choice set<\/em> for X containing exactly one member from each<\/em> element of X. <\/dd>\n<\/dl>\n

There we are. That’s it: set theory in a nutshell. You can derive pretty much all of mathematics from those 10 axioms, plus simple first order predicate logic. The integers fall out pretty naturally from the axiom of infinity; once you’ve got the integers, you can use the axiom of pairing to create the rationals; once you’ve got the rationals, you can use these axioms to derive Dedekind cuts to get the reals; once you’ve got the reals, you can use the axiom of replacement to get the transfinites. It just all flows out from these 10 rules.<\/p>\n

The amazing thing is that they’re not even particularly hard – the full meanings of some of them should be obvious already; and the others, you’ll see when we get to them that it doesn’t take too much to really understand them. It took some real genius to derive<\/em> these rules; figuring out how to draw down the entirety of set theory into 10 rules, while preventing problems like Russell’s paradox is an astonishingly difficult task. But once a couple of geniuses did that for us, the rest of us dummies are in great shape: we don’t need to be able to derive them; we just need to understand them.<\/p>\n","protected":false},"excerpt":{"rendered":"

Axiomatic set theory builds up set theory from a set of fundamental initial rules. The most common axiomatization, which we’ll be used, is the ZFC system: Zermelo-Fraenkel with choice set theory. The ZFC axiomatization consists of 8 basic rules which are pretty much universally accepted, and two rules that are somewhat controversial – most particularly […]<\/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-424","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-6Q","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/424","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=424"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/424\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=424"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=424"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}