I get a fair bit of mail from crackpots. The category that I find most annoying is the Cantor cranks. Over and over and over again, these losers send me their “proofs”.
What bugs me so much about this is how shallowly wrong they are.
What Cantor did was remarkably elegant. He showed that given anything that is claimed to be a one-to-one mapping between the set of integers and the set of real numbers (also sometimes described as an enumeration of the real numbers – the two terms are functionally equivalent), then here’s a simple procedure which will produce a real number that isn’t in included in that mapping – which shows that the mapping isn’t one-to-one.
The problem with the run-of-the-mill Cantor crank is that they never even try to actually address Cantor’s proof. They just say “look, here’s a mapping that works!”
So the entire disproof of their “refutation” of Cantor’s proof is… Cantor’s proof. They completely ignore the thing that they’re claiming to disprove.
I got another one of these this morning. It’s particularly annoying because he makes the same mistake as just about every other Cantor crank – but he also specifically points to one of my old posts where I rant about people who make exactly the same mistake as him.
To add insult to injury, the twit insisted on sending me PDF – and not just a PDF, but a bitmapped PDF – meaning that I can’t even copy text out of it. So I can’t give you a link; I’m not going to waste Scientopia’s bandwidth by putting it here for download; and I’m not going to re-type his complete text. But I’ll explain, in my own compact form, what he did.
It’s an old trick; for example, it’s ultimately not that different from what John Gabriel did. The only real novelty is that he does it in binary – which isn’t much of a novelty. This author calls it the “mirror method”. The idea is, in one column, write a list of the integers greater than 0. In the opposite column, write the mirror of that number, with the decimal (or, technically, binary) point in front of it:
Extend that out to infinity, and, according to the author, the second column it’s a sequence of every possible real number, and the table is a complete mapping.
The problem is, it doesn’t work, for a remarkably simple reason.
There is no such thing as an integer whose representation requires an infinite number of digits. For every possible integer, its representation in binary has a fixed number of bits: for any integer N, it’s representation is no longer that . That’s always a finite integer.
But… we know that the set of real numbers includes numbers whose representation is infinitely long. so this enumeration won’t include them. Where does the square root of two fall in this list? It doesn’t: it can’t be written as a finite string in binary. Where is π? It’s nowhere; there’s no finite representation of π in binary.
The author claims that the novel property of his method is:
Cantor proved the impossibility of both our enumerations as follows: for any given enumeration like ours Cantor proposed his famous diagonal method to build the contra-sample, i.e., an element which is quasi omitted in this enumeration. Before now, everyone agreed that this element was really omitted as he couldn’t tell the ordinal number of this element in the give enumeration: now he can. So Cantor’s contra-sample doesn’t work.
This is, to put it mildly, bullshit.
First of all – he pretends that he’s actually addressing Cantor’s proof – only he really isn’t. Remember – what Cantor’s proof did was show you that, given any purported enumeration of the real numbers, that you could construct a real number that isn’t in that enumeration. So what our intrepid author did was say “Yeah, so, if you do Cantor’s procedure, and produce a number which isn’t in my enumeration, then I’ll tell you where that number actually occurred in our mapping. So Cantor is wrong.”
But that doesn’t actually address Cantor. Cantor’s construction specifically shows that the number it constructs can’t be in the enumeration – because the procedure specifically guarantees that it differs from every number in the enumeration in at least one digit. So it can’t be in the enumeration. If you can’t show a logical problem with Cantor’s construction, then any argument like the authors is, simply, a priori rubbish. It’s just handwaving.
But as I mentioned earlier, there’s an even deeper problem. Cantor’s method produces a number which has an infinitely long representation. So the earlier problem – that all integers have a finite representation – means that you don’t even need to resort to anything as complicated as Cantor to defeat this. If your enumeration doesn’t include any infinitely long fractional values, then it’s absolutely trivial to produce values that aren’t included: 1/3, 1/7, 1/9.
In short: stupid, dull, pointless; absolutely typical Cantor crankery.