Another chaos theory post is in progress. But while I was working on it, a couple of
\ncomments arrived on some old posts. In general, I’d reply on those posts if I thought
\nit was worth it. But the two comments are interesting not because they actually lend
\nanything to the discussion to which they are attached, but because they are perfect
\ndemonstrations of two of the most common forms of crackpottery – what I call the
\n“Education? I don’t need no stinkin’ education” school, and the “I’m so smart that I don’t
\neven need to read your arguments” school.<\/p>\n
<\/p>\n
Let’s start with the willful ignorance. This is the kind of crackpottery \nI can’t believe there are still people who believe in Cantor’s That argument reduces, roughly, to “I have no idea what Cantor’s argument was, Cantor didn’t treat infinity like a finite number. What he did was study the The rough sketch is: take any set, S. You can create another set, called the power set<\/em> of S Some of that actually has an effect in reality. I can create a perfect one Intuitively, it seems wrong<\/em> that there are degrees of infinity, that there Moving on, we come to the genius who doesn’t need to know an argument in
\nthat frankly bugs me most. As an American, I’m used to the fact that our
\nculture distrusts intelligence and education. Politicians in America use
\n“intellectual” as a insult. Mentioning that someone attended one of the best
\nschools in America is commonly used as a criticism. That’s where this one, by
\na guy who calls himself “Vorlath”, comes from. Enough introduction, here
\nit is<\/a>:<\/p>\n
\ntheory. It’s complete silliness and makes the people who believe in
\nsuperstition look like the smart ones by comparison. Cantor was well known to
\ntreat infinity as a finite number and that’s all he’s doing with his theory.
\nIt doesn’t mean anything.\n<\/p><\/blockquote>\n
\nbut I don’t like it, and therefore it’s wrong”.<\/p>\n
\nstructure of numbers, and realize that “infinity” isn’t a simple thing. You can show
\nthat there are “infinities” that are larger than other “infinities”. In fact, it’s
\npretty inescapable.<\/p>\n
\n(usually written 2S<\/sup>), which consists of the set of all subsets of S. So if S is
\nthe empty set, then 2S<\/sup> has one value: the set containing the empty set. If S
\nis the set { a, b }, then 2S<\/sup> = { {}, {a}, {b}, {a,b} }. The power set of any set S is
\nalways<\/em> larger than S. So – take the set of all natural numbers. How big is it? Well,
\nit’s infinite. How big is its powerset? It’s also infinite. But every powerset must<\/em> be
\nbigger than the original set – so the powerset of the natural numbers must be larger
\nthan the natural numbers. How can that be? It can be, because there are different kinds
\nof infinities.<\/p>\n
\nto one mapping between the natural numbers, and the set of all rational
\nnumbers. They’re the same size. But I can’t<\/em> do that for real numbers. My
\nmapping for the rationals won’t include π, unless I cheat and specifically add it
\nto the list. It won’t include square roots. No matter what I do, I can’t devise any
\nmethod for creating a one-to-one correspondence between the natural numbers and the real
\nnumbers. That<\/em> is what Cantor’s work really showed.<\/p>\n
\nare bigger infinities and smaller infinities. The argument from Vorlath is, basically,
\nthat the fact that it’s not intuitive means that it must<\/em> be wrong. Vorlath has
\nno need to know what Cantor really said, or what it really means. Without knowing that,
\nhe just knows<\/em> it’s wrong, because it’s obviously<\/em> silly.<\/p>\n
\norder to disprove it. This is an amazingly common form of argument. I see it
\nmostly in the form of Cantor disproofs, where it takes the form “Here’s a
\none-to-one mapping between the natural numbers and the reals”. The mappings
\nare always wrong. But what makes the argument particularly annoying is that
\nthe mappings are trivially<\/em> wrong: Cantor’s diagonalization shows that
\ngiven any<\/em> one-to-one mapping between the naturals and the reals, the
\nmapping will miss some real numbers. In every<\/em> case where I’ve seen
\none of these arguments, their enumerated mapping fails because Cantor’s
\ndiagonalization shows how to produce a counterexample for it. You can’t disprove
\nCantor’s diagonalization without showing that something is wrong with the
\ndiagonalization proof – but these anti-Cantor geniuses constantly think that
\nthey can disprove Cantor without actually addressing the proof.<\/p>\n