{"id":2212,"date":"2013-07-29T11:00:19","date_gmt":"2013-07-29T15:00:19","guid":{"rendered":"http:\/\/scientopia.org\/blogs\/goodmath\/?p=2212"},"modified":"2013-07-29T11:00:19","modified_gmt":"2013-07-29T15:00:19","slug":"infinite-cantor-crankery","status":"publish","type":"post","link":"http:\/\/www.goodmath.org\/blog\/2013\/07\/29\/infinite-cantor-crankery\/","title":{"rendered":"Infinite Cantor Crankery"},"content":{"rendered":"<p> I recently got <em>yet another<\/em> email from a Cantor crank.\n<\/p>\n<p> Sadly, it&#8217;s not a particularly interesting letter.  It contains an argument that I&#8217;ve seen more times than I can count. But I realized that I don&#8217;t think I&#8217;ve ever written about this particular boneheaded nonsense!<\/p>\n<p> I&#8217;m going to paraphrase the argument: the original is written in broken english and is hard to follow.<\/p>\n<blockquote>\n<ul>\n<li> Cantor&#8217;s diagonalization creates a magical number (&#8220;Cantor&#8217;s number&#8221;) based on an infinitely long table.<\/li>\n<li> Each digit of Cantor&#8217;s number is taken from one row of the table: the <em>N<\/em>th digit is produced by the <em>N<\/em>th row of the table.<\/li>\n<li> This means that the <em>N<\/em>th digit only exists after processing <em>N<\/em> rows of the table.<\/li>\n<li> Suppose it takes time <em>t<\/em> to get the value of a digit from a row of the table.\n<li> Therefore, for any natural number <em>N<\/em>, it takes <em>N*t<\/em> time to get the first <em>N<\/em> digits of Cantor&#8217;s number.<\/li>\n<li> Any finite prefix of Cantor&#8217;s number is a rational number, which is clearly in the table.<\/li>\n<li> The full Cantor&#8217;s number doesn&#8217;t exist until an <em>infinite<\/em> number of steps has been completed, at time <im>&infinity;*t<\/im>.<\/li>\n<li> Therefore Cantor&#8217;s number <em>never exists<\/em>. Only finite prefixes of it exist, and they are all rational numbers.<\/li>\n<\/ul>\n<\/blockquote>\n<p> The problem with this is quite simple: Cantor&#8217;s proof doesn&#8217;t <em>create<\/em> a number; it <em>identifies<\/em> a number.<\/p>\n<p> It might take an infinite amount of time to figure out which number we&#8217;re talking about &#8211; but that doesn&#8217;t matter. The number, like all numbers, exists, independent of<br \/>\nour ability to compute it. Once you accept the rules of real numbers as a mathematical framework, then all of the numbers, every possible one, whether we can identify it, or describe it, or write it down &#8211; they all exist. What a mechanism like Cantor&#8217;s diagonalization does is just give us a way of identifying a particular number that we&#8217;re interested in. But that number exists, whether we describe it or identify it.<\/p>\n<p> The easiest way to show the problem here is to think of other irrational numbers. No irrational number can ever be written down completely. We know that there&#8217;s got to be some number which, multiplied by itself, equals 2. But we can&#8217;t actually write down all of the digits of that number. We can write down progressively better approximations, but we&#8217;ll never actually write the square root of two. By the argument above against Cantor&#8217;s number, we can show that the square root of two doesn&#8217;t exist. If we need to <em>create<\/em> the number by writing down all af its digits,s then the square root of two will <em>never<\/em> get created! Nor will any other irrational number. If you insist on writing numbers down in decimal form, then neither will many fractions. But in math, we don&#8217;t create numbers: we describe numbers that already exist.<\/p>\n<p> But we could weasel around that, and create an alternative formulation of mathematics in which all numbers must be writeable in some finite form. We wouldn&#8217;t need to say that we can create numbers, but we could constrain our definitions to get rid of the nasty numbers that make things confusing. We could make a reasonable argument that those problematic real numbers don&#8217;t really exist &#8211; that they&#8217;re an artifact of a flaw in our logical definition of real numbers. (In fact, some mathematicians like Greg Chaitin have actually made that argument semi-seriously.)<\/p>\n<p> By doing that, irrational numbers could be defined out of existence, because they<br \/>\ncan&#8217;t be written down. In essence, that&#8217;s what my correspondant is proposing: that the definition of real numbers is broken, and that the problem with Cantor&#8217;s proof is that it&#8217;s based on that faulty definition. (I don&#8217;t think that he&#8217;d agree that that&#8217;s what he&#8217;s arguing &#8211; but either numbers exist that can&#8217;t be written in a finite amount of time, or they don&#8217;t. If they do, then his argument is worthless.)<\/p>\n<p> You certainly <em>can<\/em> argue that the only numbers that should exist are numbers that can be written down. If you do that, there are two main paths. There&#8217;s the theory of computable numbers (which allows you to keep &pi; and the square roots), and there&#8217;s the theory of rational numbers (which discards everything that can&#8217;t be written as a finite fraction). There are interesting theories that build on either of those two approaches. In both, Cantor&#8217;s argument doesn&#8217;t apply, because in both, you&#8217;ve restricted the set of numbers to be a countable set.<\/p>\n<p> But that doesn&#8217;t say anything about the theory of real numbers, which is what Cantor&#8217;s proof is talking about. In the real numbers, numbers that can&#8217;t be written down in any form do exist. Numbers like the number produced by Cantor&#8217;s diagonalization definitely do. The infinite time argument is a load of rubbish because it&#8217;s based on the faulty concept that Cantor&#8217;s number doesn&#8217;t exist until we create it.<\/p>\n<p> The interesting thing about this argument to be, is its selectivity. To my correspondant, the existence of an infinitely long table isn&#8217;t a problem. He doesn&#8217;t think that there&#8217;s anything wrong with the idea of an infinite process creating an infinite table containing a mapping between the natural numbers and the real numbers. He just has a problem with the infinite process of traversing that table. Which is really pretty silly when you think about it.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>I recently got yet another email from a Cantor crank. Sadly, it&#8217;s not a particularly interesting letter. It contains an argument that I&#8217;ve seen more times than I can count. But I realized that I don&#8217;t think I&#8217;ve ever written about this particular boneheaded nonsense! I&#8217;m going to paraphrase the argument: the original is written [&hellip;]<\/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":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[2,11],"tags":[],"class_list":["post-2212","post","type-post","status-publish","format-standard","hentry","category-bad-math","category-cantor-crankery"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-zG","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/2212","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=2212"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/2212\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=2212"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=2212"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=2212"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}