Today, I’m going to try to show you an example of why the axiom makes so many people so uncomfortable. When you get down to the blood and guts of what it means, it implies some *very* strange things. What I’m going to do today is tell you about one of those: the Banach-Tarski paradox, in which you can create two spheres of size S out of one sphere of size S cutting the single sphere into pieces, and then gluing those pieces back together. Volume from nowhere, and your spheres for free!<\/p>\n
Today, I’m going to try to show you an example of why the axiom makes so many people so uncomfortable. When you get down to the blood and guts of what it means, it implies some *very* strange things. What I’m going to do today is tell you about one of those: the Banach-Tarski paradox, […]<\/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-436","post","type-post","status-publish","format-standard","hentry","category-set-theory"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-72","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/436","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=436"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/436\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=436"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=436"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=436"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nAs I said in the post on the axiom of choice, it was very controversial at first: controversial enough that many people made attempts to disprove it. The most famous of those attempts was developed by two mathematicians: Stefan Banach and Alfred Tarski, based on the work of Hausdorff. They produced a result which goes beyond unintuitive and into the realm of downright strange. Their result is now known as the Banach-Tarski paradox (although strictly speaking, it isn’t a paradox), and while it failed as a disproof of the axiom of choice, it’s now commonly used as the<\/em> example of why the axiom of choice is weird.
\nWhat the BT paradox says is that it’s possible to take a sphere, cut it into a small
\nfinite number of pieces, and to reassemble those pieces into two spheres the same size as the original. This involves *no* stretching or deformation of the pieces at all: just translation, and rotation. There are *no* gaps in the two spheres. No copying of pieces. No clever tricks *at all* in how you glue the pieces together to form new spheres – and yet the volume doubles. You’ve gotten 2 spheres from one.
\nHow does it work? Of course there’s a trick to it. Banach and Tarski *thought* that the trick should serve as a disproof of the axiom of choice – so it’s not a simple trick.
\nThe trick, such as it is, involves the *shape* of the pieces. The shapes consist of *unmeasurable* sets of points: intuitively, their shapes are so complex that you can’t describe them accurately even using a countably infinite number of points. So it’s not something that you can actually do with a model of a sphere: these “shapes” have no measurable volume (in the sense that their volume and surface area isn’t definable), and their edges aren’t finite.
\nBut there’s a formulation of a constraint in terms of sets where the axiom of choice says that these sets *exist*, even if we can’t find them.
\nThe proof of it goes way beyond what I want to write here. The basic idea of it is that there’s a geometric notion of equidecomposablility<\/em>: two objects A and B are equidecomposable if and only if they can be broken into sets A=∪i=1,n<\/sub>ai<\/sub> and B=∪b=1,j<\/sub>bj<\/sub> where there’s a one to one mapping between the ai<\/sub> and bj<\/sub> such that ai<\/sub> maps to bj<\/sub> if and only if ai<\/sub> is geometrically congruent with bj<\/sub>.
\nThrough some nifty trickery with group theory and the AC, you can then show that two spheres are equidecomposable with a single sphere *minus* the center point; and then separately show that the sphere minus the missing point is equidecomposable with the sphere. The whole thing works off of choice by choosing elements from the powerset of points in the sphere: using choice, you can infer the existence of bizarre
\nnonconstructable sets that happen to meet the requirements.
\nIf the idea of creating phantom volume out of nowhere through the axiom of choice doesn’t make it clear to you why people are bothered by it, then I don’t know what will.<\/p>\n","protected":false},"excerpt":{"rendered":"