{"id":429,"date":"2007-05-24T21:43:12","date_gmt":"2007-05-24T21:43:12","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/2007\/05\/24\/the-axiom-of-infinity\/"},"modified":"2007-05-24T21:43:12","modified_gmt":"2007-05-24T21:43:12","slug":"the-axiom-of-infinity","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2007\/05\/24\/the-axiom-of-infinity\/","title":{"rendered":"The Axiom of Infinity"},"content":{"rendered":"

The axiom of infinity is a bundle of tricks. As I said originally, it does two things. First, it gives us our first infinite set; and second, it sets the stage for representing arithmetic in terms of sets. With the axiom of infinity, we get the natural numbers; with the natural numbers, we can get the integers; with the integers, we can get the rationals. Once we have the rationals, things get a bit harder – but we can get the reals via Dedekind cuts; and by transfinite induction, we can get the transfinite numbers. But before we can get to any of that, we need a sound representation of the naturals in terms of sets.<\/p>\n

<\/p>\n

Let’s look at the statement of the axiom again:<\/p>\n

∃N : ∅∈N ∧ (∀x:x∈N ⇒ (x∪{x})∈N).<\/p>\n

That’s a classic recursive definition, consisting of a base case and an inductive case. The base tells us the initial value of N – the empty set. And the inductive case shows us how we can construct the further members of N.<\/p>\n

Walk through a few steps of that:<\/p>\n

    \n
  1. By the base case ∅ ∈ N.<\/li>\n
  2. Since ∅∈N, then ∅∪{∅}={∅}∈N.<\/li>\n
  3. Since {∅}∈N, then {∅}∪{{∅}}={∅,{∅}}∈N.<\/li>\n
  4. Since {∅,{∅}}∈N, then {∅,{∅}}∪{{∅,{∅}}}={∅,{∅},{∅,{∅}}}∈N<\/li>\n<\/ol>\n

    So we’re looking at an infinite sequence of values, which have a peculiar property. The N+1th value in the series consists of the set of the first N values – the set of all values that preceeded it.<\/p>\n

    If you look at this series through the lens of the Peano
    \naxioms<\/a>, you can see that it works for the integers. ∅ is the initial value, 0. For any number i, there’s a unique successor {i}∪{{i}}. Every member of the set is the successor to exactly one number – the larger number which is a subset<\/em> of it. There’s only one possible successor to any number. And while it’s less obvious than the others, you can, with the help of some of the other axioms, work through the induction rule, showing that it works. So the axiom of infinity is giving us a set-based view of
    \nthe natural numbers.<\/p>\n

    But we know that the natural numbers aren’t enough to do all of math, right? We’ve seen in other places that the set of natural numbers isn’t large enough – a construction that gives us natural numbers can’t be used to represent all of the reals, because no matter how we do it, the set of naturals is just too small.<\/p>\n

    That’s where the powerset axiom is going to come in. The powerset – the set of all subsets of this infinite set – is larger than this. And the powerset of that set is still larger. We’ll see more about that later. But first, we’re going to have to start being a lot more precise. So far, we’ve been able to mostly just gloss over the distinction between sets and classes, because everything we’ve talked about so far is a set. But when we start getting into some of the really big sets – sets bigger than the reals – then things start getting a bit crazy, and the set\/proper class distinction starts to become very important.<\/p>\n","protected":false},"excerpt":{"rendered":"

    The axiom of infinity is a bundle of tricks. As I said originally, it does two things. First, it gives us our first infinite set; and second, it sets the stage for representing arithmetic in terms of sets. With the axiom of infinity, we get the natural numbers; with the natural numbers, we can get […]<\/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-429","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-6V","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/429","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=429"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/429\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=429"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=429"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=429"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}