G\u00f6del’s incompleteness theorem is an interesting little piece of mathematics but probably gets talked about more than it really deserves to be. I’m going to add to the problem here by talking about it.<\/p>\n
It’s a pretty simple idea really – if you have a formal system of mathematics F sufficiently rich to do anything interesting, you can write down a statement G that says, essentially, “statement G cannot be proved in formal system F”. G must be true since if it was false it would be provable and therefore true, which is a contradiction. So given any formal system you can either use it to prove a contradiction (in which case it can be used to be prove anything, and is therefore useless) or you can find a true statement that cannot be proved within the formal system. Mathematics (being a non-contradictory formal system) can therefore never be used to prove all the statements that are true. “But hang on a minute,” I hear you say, “that proof that G is true looked an awful lot like mathematics to me.” Really all this means is that the kind of things we think about as being mathematical proofs can’t be enumerated into some nice tidy finite set, since whatever you put in that set you can use to find another element that needs to be added.<\/p>\n
There is an analogy here with Cantor’s diagonal proof that the real numbers form a greater class of infinity than the integers. While they’re both interesting proofs there is a fundamental “infinite-ness” and “strange loop-iness” about both of them. Infinite things are strange and non-intuitive, and also useless for describing the real universe (since we can never measure anything with infinite precision).<\/p>\n","protected":false},"excerpt":{"rendered":"
G\u00f6del’s incompleteness theorem is an interesting little piece of mathematics but probably gets talked about more than it really deserves to be. I’m going to add to the problem here by talking about it. It’s a pretty simple idea really – if you have a formal system of mathematics F sufficiently rich to do anything […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[5],"tags":[],"class_list":["post-649","post","type-post","status-publish","format-standard","hentry","category-maths"],"_links":{"self":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/649","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/comments?post=649"}],"version-history":[{"count":3,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/649\/revisions"}],"predecessor-version":[{"id":801,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/posts\/649\/revisions\/801"}],"wp:attachment":[{"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/media?parent=649"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/categories?post=649"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.reenigne.org\/blog\/wp-json\/wp\/v2\/tags?post=649"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}