Some of the basic axioms of ZFC set theory can seem a bit uninteresting on their own. But when you take them together, and reason your way around them, you can find some interesting things.<\/p>\n
Let’s start by looking at the axiom of extensionality. Pretty simple, right? All it does is define what set equality means. It says that two sets are equal if, and only if, they have the same members: that is, a set is completely determined by its contents.<\/p>\n
How much more trivial can a statement about sets get? It really doesn’t seem to say much. But what happens when we start thinking through what that means?<\/p>\n
The way we normally think of sets, they’re collections of objects. So, imagine a set like {red, green, blue}, where the three values are atoms<\/em>: that is, they’re single objects, not collections. What does the axiom of extensionality say about that? It says that red, green, and blue are not<\/em> atoms?<\/p>\n Why? Well – let’s look at the axiom of extensionality again: (∀A,B: A=B ⇔ In fact, if we follow that reasoning through, there’s only one possible atom: the only set with no members is the empty set. So anything else we want, we’re going to have to represent using some kind of collection.<\/p>\n As a result of that, along with the axiom of specification, we can show that the axiom Some of the basic axioms of ZFC set theory can seem a bit uninteresting on their own. But when you take them together, and reason your way around them, you can find some interesting things. Let’s start by looking at the axiom of extensionality. Pretty simple, right? All it does is define what set equality […]<\/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-427","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-6T","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/427","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=427"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/427\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=427"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=427"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=427"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\n(∀C: C∈A ⇔ C∈B)). So – does red = blue? Well, if they’re atoms, then
\nyes, red=blue, because nothing is in red, and nothing is in blue. Since neither has any members, they’re equal.<\/p>\n
\nof the empty set is actually redundant. After all – the axiom of specification basically
\nsays that if you can describe a collection of values by a predicate, that collection is a class. So take the predicate P(x)=false; that’s a set with no values. Also known as the empty set. So the empty set exists, and it’s the only set with no members – aka the only atom.<\/p>\n","protected":false},"excerpt":{"rendered":"