{"id":447,"date":"2007-06-21T18:44:46","date_gmt":"2007-06-21T18:44:46","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/06\/21\/alternative-axioms-nbg-set-theory\/"},"modified":"2014-08-15T10:26:50","modified_gmt":"2014-08-15T14:26:50","slug":"alternative-axioms-nbg-set-theory","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/06\/21\/alternative-axioms-nbg-set-theory\/","title":{"rendered":"Alternative Axioms: NBG Set Theory"},"content":{"rendered":"

So far, we’ve been talking mainly about the ZFC axiomatization of set theory, but in fact, when I’ve talked about classes, I’ve really been talking about the von Newmann-Bernays-Gödel definition of classes. (For example, the proof I showed the other day that the ordinals are a proper class<\/a> is an NBG proof.) NBG is an alternate formulation of set theory which has the same proof power as ZFC, but does it with a finite set of axioms. (If you recall, several of the axioms of ZFC are actually axiom schemas<\/em>, which need to be distinctly instantiated for all possible predicates.) NBG uses one axiom scheme, but it’s possible to show that that schema only expands into a finite number of distinct axioms.<\/p>\n

<\/p>\n

In ZFC, the sets are the things you can construct using the axioms; classes are things that are sets in naive set theory, but which you can’t define properly using the ZFC axioms. ZFC doesn’t talk about classes. NBG actually defines and uses classes as its basis, and it’s where the set\/proper class distinction comes from: a class is a collection of sets; and a set is a class that’s a member of some other class.<\/p>\n

Since NBG has both sets and classes, it’s effectively a typed theory, where there are two types: sets, and classes. We’ll use lowercase letters for sets, and uppercase for classes – and we’ll use that for axioms as well as variables (there’s an “Axiom of Extensionality” for classes, and an “axiom of extensionality” for sets).<\/p>\n

For NBG, we define the correspondence betweens sets and classes with the same members using an onto function Rep<\/em>, where Rep(A)=a if ∀x: x∈a⇔x∈A.<\/p>\n

The axioms of NBG are:<\/p>\n

    \n
  1. The axiom of class extensionality<\/b>: (∀x: x;∈A ⇔ x∈B ⇒ A=B): classes are equal if and only if they have the same members.<\/li>\n
  2. The axiom of set extensionality<\/b>: (∀x: x;∈a ⇔ x∈x ⇒ a=b): sets are equal if and only if they have the same members.<\/p>\n
  3. The axiom of Class Comprehension<\/b>: For any formula \\phi which does not<\/em> quantify over classes, there is a class C such that x \\in C if and only if \\phi(x) is true. (This is actually a schema parametric in \\phi; but there’s an equivalent way of expanding this into a finite number of axioms. Trust me. Or if enough people ask, I’ll show you how the expansion works.)<\/li>\n
  4. The Axiom of Pairing<\/b>: \\forall x,y: \\exists ;: z=\\{x, y\\}: for any two sets x and y, there’s a set with exactly x and y as members.<\/li>\n
  5. The Axiom of size<\/b>: For any class C, there is a set c such that c=\\text{Rep}(C) if and only if there is no total one-to-one function between C and the class of all sets. This is the one that creates that definition of classes as things to big to be sets.<\/li>\n
  6. The Axiom of union<\/b>: \\forall x: (\\exists y: \\forall a \\in x: \\forall b \\in a: b \\in y) for any set x, there is a set containing the members of the members of x. In other words, you can take the union of any set of sets.<\/li>\n
  7. the Axiom of powerset<\/b>: \\forall x: (\\exists y = \\{ a \\subseteq x \\}): For any set x, there is a set which contains all of the subsets of x.<\/li>\n
  8. The Axiom of infinity<\/b>: There exists a set N where:\n
      \n
    • The empty set ∅ is a member of N; and;<\/li>\n
    • for each member x∈N, (x∪{x})∈N<\/li>\n<\/ul>\n

      Which is a fancy way of saying that there’s at least one infinitely large set, which is exactly Cantor’s construction of the natural numbers.\n<\/li>\n

    • The Axiom of Foundation<\/b>: All non-empty classes are disjoint from at least one of their elements.<\/li>\n
    • The Axiom of regularity<\/b>: All non-empty sets are disjoint from at least one of their elements.<\/li>\n<\/ol>\n

      Personally, I like NBG better that ZFC, precisely because it provides ways of talking about classes; ZFC pretty much throws up its hands and says “Outside my realm” when you get classes. I think that capturing proper classes in your set theory is both cool and useful. I also find proofs based an the axiom of size to be clearer than most other ways of proving things to be proper classes. Unfortunately for me, ZFC is pretty dominant in mathematical circles. Personally, I don’t really see why, but that’s the way it is.<\/p>\n","protected":false},"excerpt":{"rendered":"

      So far, we’ve been talking mainly about the ZFC axiomatization of set theory, but in fact, when I’ve talked about classes, I’ve really been talking about the von Newmann-Bernays-Gödel definition of classes. (For example, the proof I showed the other day that the ordinals are a proper class is an NBG proof.) NBG is an […]<\/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-447","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-7d","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/447","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=447"}],"version-history":[{"count":3,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/447\/revisions"}],"predecessor-version":[{"id":3024,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/447\/revisions\/3024"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=447"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=447"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=447"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}