Sets are truly amazing things. In the history of mathematics, they’re
\na remarkably recent invention – and yet, they’re now considered to be the
\nfundamental basis on which virtually all of mathematics is built. From simple things (like the natural numbers), to the most abstract and esoteric things (like algebras, or topologies, or categories), in modern math, they’re pretty much all understood
\nin terms of sets.<\/p>\n
<\/p>\n
So what is a set? A set is really just an abstract way about talking about a Sets don’t<\/em> have any intrinsic concept of ordering – that is, you can’t talk about the “first” thing in a set, or which of two things in a set comes first. Doing that involves creating an operation on the members of the set – and then you’re talking about the ordering properties of the operation<\/em>, not of the set itself.<\/p>\n When we write sets, we typically write them as a list of members surrounded by curly braces: {1, 2, 3}, {“bob”, “joe”, “mike”, “sam”}, etc. The set with no members, Commonly, we talk about a very small number of basic fundamental operations that we can talk about for sets:<\/p>\n So, why is this stuff so fundamental? What’s so powerful about this? Let’s Within this construction, N<M if and only if N∈M. And if you take the time The main problem with this version of set theory is that it’s very easy to The existence of things like this in naive set theory were considered unacceptable, a sign that there was something wrong with the construction of set theory that needed to be fixed. A lot of effort went into creating more structured versions of set theory that tried to avoid paradoxes, but Gödel ultimately showed that no matter how much you did to restrict it, you were trapped – either you wind up with a theory that is incomplete (includes true statements that can’t be proven) or inconsistent (contains paradoxical statements). But that is definitely not<\/em> a subject for a basics post!<\/p>\n","protected":false},"excerpt":{"rendered":" Sets are truly amazing things. In the history of mathematics, they’re a remarkably recent invention – and yet, they’re now considered to be the fundamental basis on which virtually all of mathematics is built. From simple things (like the natural numbers), to the most abstract and esoteric things (like algebras, or topologies, or categories), in […]<\/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":[74],"tags":[],"class_list":["post-295","post","type-post","status-publish","format-standard","hentry","category-basics"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4L","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/295","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=295"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/295\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=295"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=295"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=295"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\ncollection of distinct things. Really, in the simplest version of set theory,
\nthat’s it<\/em>. Such a simple notion! And yet – from there, the entire world of math opens up. (As an aside, in the rest of this article, I’m going to be talking about simple set theory, or naive set theory; there is a more advanced variant that was part of the attempt to create a more restrictive version of set theory that avoided paradoxical statements.)<\/p>\n
\ncalled the empty set is written either {}, or ∅. We also often use a notation called a set comprehension<\/em>, which specifies the members of a set in terms of some property. {x | x is a friend of mine}, {x | x < 200}, etc.<\/p>\n\n
\ncontaining everything that is in either A or B – that is, {x | x ∈ A or<\/em> x ∈ B}<\/li>\n
\nof the elements of A that are not<\/em> also elements of B; that is,
\n{x : x∈A and<\/em> x∉B}.<\/li>\n
\nthe set consisting of the union<\/em> of AB and BA: {x : (x∈A and<\/em> x∉B) or<\/em> (x∉A and<\/em> x∈B)}.<\/li>\n<\/ol>\n
\nlook at one simple example. We saw the natural numbers the other day – let’s look at
\nhow to define the natural numbers using sets:<\/p>\n\n
\nto work it through, you can see that this construction satisfies the Peano axioms –
\nthese are<\/em> the integers.<\/p>\n
\ncreate paradoxical statements. Since we can have sets that contain sets, and there are no restrictions, then we can create all kinds of strange structures. The classic example is: X={S | S∉S } – that is, the set of all sets that are not members of
\nthemselves. It’s a simple paradox: is X∈X? If X is<\/em> a member of itself, then by the definition of X, it’s not a member of itself; but if it’s not a member of itself, then by its definition, it must<\/em> be a member of itself!<\/p>\n