Enough preliminaries. Let’s dive in and see what he did. His abstract
\ngives about a coherent a description as anything else in the paper, so
\nwe’ll start with that.<\/p>\n
\nCantor’s diagonal argument makes use of a hypothetical table
\nT containing all real numbers within the real interval (0,1). That table
\ncan be easily rede\ufb01ned in order to ensure it contains at least all rational
\nnumbers within (0,1). In these conditions, could the rows of T be reordered
\nso that the resulting diagonal and antidiagonal were rational numbers? In
\nthat case not only the set of real numbers but also, and for the same reason,
\nthe set of rational numbers would be nondenumerable. And then we would
\nhave a contradiction since Cantor also proved the set of rational numbers is
\ndenumerable. Should, therefore, Cantor’s diagonal argument be suspended
\nuntil it be proved the impossibility of such a reordering? Is that reordering
\npossible? This paper address both questions.\n<\/p><\/blockquote>\n
To understand this, let’s do a quick review of Cantor’s diagonalization.
\nCantor is trying to prove that the set of real numbers is strictly larger than
\nthe set of natural numbers. He uses proof by contradiction: he starts by
\nsupposing that the naturals and the reals have the same size, then shows how
\nthat inevitably leads to a contradiction. <\/p>\n
If the set of real numbers is the same size as the set of natural numbers,
\nthen there is a one-to-one mapping f from the natural numbers to the real
\nnumbers in the range (0, 1). So he uses that to lay out a table, where the
\nfirst row is f(0), the second row is f(1), etc. In the table, the first column
\nis the first digit; the second column is the second digit, and so on.<\/p>\n
If the mapping is really one-to-one, then every real number must be in the
\ntable. But Cantor shows how you can easily create a new real number which is
\nnot<\/em> in the table. All you do is look at the digit in position (1,1)
\nin the grid – and change it. Then look at the digit in position (2,2), and
\nchange that. Then the digit in (3,3). And so on: for row N in the table, you
\nchange digit #n. What that procedure does is generate a number which is
\ndifferent from every number in the table in at least one digit. Therefore it’s
\nnot in the table. That’s a contradiction: we said that every real number had to
\nbe in the table, but we’ve just constructed a real number which isn’t.<\/p>\n What our author is proposing is to take Cantor’s diagonalization,
\nand do two things to it.<\/p>\n
First, he changes it so that it’s a mapping from the natural numbers
\nto the rationals<\/em> instead of the natural numbers to the reals.<\/p>\n Then, he looks at the diagonal of the and re-arrange<\/em> the rows of it.
\nHe re-arranges the rows of the table until the number in the diagonal is
\na rational. Now he’s got a table which contains all of the rationals, and whose
\nCantor diagonal is a rational number. So it looks like he’s got a counter-example
\nfor the idea that there’s a one-to-one map between the naturals and the
\nrationals. If that were the case, then Cantor would be in real trouble: Cantor
\nalso wrote a well-known proof that there’s a one-to-one mapping between the
\nnatural numbers and the rationals. So if our intrepid author is correct, then
\neither Cantor is wrong about there being no<\/em> mapping between the
\nnaturals and the reals; or he’s wrong about there being a mapping between the
\nnaturals and the rationals; or his entire system of comparing the cardinality
\nof infinite sets is completely inconsistent.<\/p>\n Looked at naively, it seems sort of compelling: if we can build
\na Cantor table that shows that the rationals aren’t countable, then
\nCantor is wrong. So what’s wrong with this proof? <\/p>\n
Reordering.<\/p>\n
Remember: in this proof, we start with a standard Cantor diagonal over the
\nrationals. That is, we start with an enumeration of rationals, lay it out in a
\ntable, and then read off a number which isn’t in the set of rational numbers.
\nIn other words, we’ve used a Cantor table to produce an irrational number. At
\nthis point, there’s nothing remotely compelling: we know<\/em> that there
\nare irrational numbers, and all that the construction did at this level is
\ngenerate one. This is neither surprising nor particularly interesting, and
\nit’s certainly no threat to Cantor’s famous proof.<\/p>\n Once he’s got the construction of the irrational, he re-arranges<\/em>
\nthe rows of the table. He tries to re-arrange it so that the number that reads
\ndown the diagonal of the table is a rational. This is exactly the problem: he
\ncan’t<\/em> do that.<\/p>\n Why not? Because the construction of the re-ordering is invalid<\/em>. To
\nquote the paper:<\/p>\n\nIf it were possible to reorder the rows of T in such a way that a rational antidiagonal
\ncould be defined, then we would have two contradictory results: the set Q of
\nrational numbers would and would not be denumerable\n<\/p><\/blockquote>\n
That is, the re-ordering is absolutely critical to his argument. But
\nthe re-ordering is, itself, self-contradictory.<\/p>\n
The argument for the existence of the re-ordering is that
\neven irrational numbers generally have some probabilistic properties
\nabout their digits. Using those, we can define an initial table
\nwhere its counter-example number has a desired set of properties
\nin the distribution of its digits. (You could use a variety of
\nproperties – but, for example, if the distribution of digits is
\nuniform, then you could conceptually re-order the rows to produce
\n0.12345678901234567890…)<\/p>\n
So far so good. Now, you re-order the rows, so that the diagonal is
\na rational number.<\/p>\n
Here’s the problem: you’re constructing a chosen<\/em> rational
\nnumber. That is, you know<\/em> what rational number you’re re-ordering the
\nrows to create. Since it’s a rational, it’s got to be in the table. And since
\nyou know<\/em> what rational it is, you’ve got to know what row in the table
\nit’s going to be. So go look at that row. <\/p>\n By the definition of the diagonalization, the value of the diagonal must<\/em>
\nbe different from the value of any of the rows by at least one digit. So the
\nrational number that you’re forming must be different from itself<\/em> by
\nat least one digit.<\/p>\n Bzzt. No good. The re-ordered rational diagonalization is self-contradictory.
\nIn fact, it’s a classic self-referential foulup.<\/p>\n
This is an obvious<\/em> problem, and it’s appalling that the author
\nof the paper, who is supposedly a math professor<\/em>, couldn’t see it. For all
\nof the crazy rigamarole he goes through to construct his re-ordering, he never
\nbothers to look at this simple problem. What kind of mathematician could build
\na construction like this and never consider the self-referential case?<\/p>\n I’ll give the author one thing: at least he actually addressed<\/em> Cantor’s
\nproof. Most authors never bother to do that. Still, he doesn’t really appear to understand
\nthe way that it works – else he’d have have noticed the self-reference problem.<\/p>\n Back at the beginning of the post, I said he almost<\/em> avoids the usual problem of ignoring the diagonalization. The catch is that, as we’ve seen above,
\nhe got it wrong<\/em>, because he didn’t remember to consider the key
\nproperty of the diagonalization: that it’s different<\/em> from every row in the
\ntable. By trying to construct a diagonal that is equal to<\/em> a row in the table,
\nhe’s doing something self-contradictory. But he ignores that property – and then when
\ndoing something self-contradictory results in a contradiction, he tries to claim that
\nit shows that one of history’s most profound and important mathematical results is
\nwrong.<\/p>\n Bozo.<\/p>\n","protected":false},"excerpt":{"rendered":"
Poor Georg Cantor. During his life, he suffered from dreadful depression. He was mocked by his mathematical colleagues, who didn’t understand his work. And after his death, he’s become the number one target of mathematical crackpots. As I’ve mentioned before, I get a lot of messages either from or about Cantor cranks. I could easily […]<\/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":[11],"tags":[],"class_list":["post-841","post","type-post","status-publish","format-standard","hentry","category-cantor-crankery"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-dz","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/841","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=841"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/841\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=841"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=841"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=841"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}