Over the weekend, a reader sent me links to not one but two new Cantor cranks!
Sadly, one of them is the incoherent sort – the kind of nutjob who strings together words in meaningless ways. Without a certain minimal rationality, there’s nothing I can say. What I try to do on this blog isn’t just make fun of crackpots – it’s explain what they get wrong! If a crackpot strings together words randomly, and no one can make any sense of just what the heck they’re saying, there’s no way to do that.
On the other hand, the second guy is a whole different matter. He’s making a very common mistake, and he’s making it very clearly. So for him, it’s well worth taking a moment and looking at what he gets wrong.
My mantra on this blog has always been: “the worst math is no math”. This is a perfect example.
First, I believe that Cantor derived a false conclusion from the diagonal method.
I believe that the primary error in the theory is not with the assertion that the set of Real Numbers is a “different size” than the set of Integers. The problem lies with the assertion that the set of Rational Numbers is the “same size” as the set of Integers. Our finite notion of size just doesn’t extend to infinite sets. Putting numbers in a list (i.e., creating a one-to-one correspondence between infinite sets) does not show that they are the same “size.”
This becomes clear if we do a two step version of the diagonal method.
Step One: Lets start with the claim: “Putting an infinite set in a list shows that it is the same size as the set of Integers.”
Step Two: Claiming to have a complete list of reals, Cantor uses the diagonal method to create a real number not yet in the list.
Please, think about this two step model. The diagonal method does not show that the rational numbers are denumerable while the real numbers are not. The diagonal method shows that the assertion in step one is false. The assertion in step one is as false for rational numbers as it is for real numbers.
The diagonal method calls into question the cross-section proof used to show that the rational numbers are the same size as the integers.
Cantor didn’t talk about size. He never asserted anything about size. He asserted something about cardinality.
That might sound like a silly nitpick: it’s just terminology, right?
Wrong. What does size mean? Size is an informal term. It’s got lots of different potential meanings. There’s a reasonable definition of “size” where the set of natural numbers is larger than the set of even natural numbers. It’s a very simple definition: given two sets of objects A and B, the size of B is larger than the size of A if A is a subset of B.
When you say the word “size”, what do you mean? Which definition?
Cantor defined a new way of defining size. It’s not the only valid measure, but it is a valid measure which is widely useful when you’re doing math. The measure he defined is called cardinality. And cardinality, by definition, says that two sets have the same cardinality if and only if it’s possible to create a one-to-one correspondance between the two sets.
When our writer said “Our finite notion of size just doesn’t extend to infinite sets”, he was absolutely correct. The problem is that he’s not doing math! The whole point of Cantor’s work on cardinality was precisely that our finite notion of size doesn’t extend to infinite sets. So he didn’t use our finite notion of size. He defined a new mathematical construct that allows us to meaningfully and consistently talk about the size of infinite sets.
Throughout his website, he builds a complex edifice of reasoning on this basis. It’s fascinating, but it’s ultimately all worthless. He’s trying to do math, only without actually doing math. If you want to try to refute something like Cantor’s diagonalization, you can’t do it with informal reasoning using words. You can only do it using math.
This gets back to a very common error that people make, over and over. Math doesn’t use fancy words and weird notations because mathematicians are trying to confuse non-mathematicians. It’s not done out of spite, or out of some desire to exclude non-mathematicians from the club. It’s about precision.
Cantor didn’t talk about the cardinality of infinite sets because he thought “cardinality” sounded cooler and more impressive than “size”. He did it because “size” is an informal concept that doesn’t work when you scale to infinite sets. He created a new concept because the informal concept doesn’t work. If you’re argument against Cantor is that his concept of cardinality is different from your informal concept of size, you’re completely missing the point.
I have another quibble with the second crank. His explanation of the Cantor’s diagonalization argument.
This isn’t quite accurate. Cantor’s Diagonalization doesn’t really rely on contradiction, as you don’t need to assume that you have a “complete list” or a bijection. First you note there is an embedding of the natural numbers into the interval (0,1), by mapping n to 1/(n+2). (Analogously you can show this for a set and it’s power set by considering the complement of each element in the base set.) Then using Cantor’s Diagonalization I show that any embedding into the reals doesn’t map to at least one element, thus there can’t be a bijection between the sets (ie they aren’t in 1-1 correspondence). So the reals have strictly larger cardinality than the natural numbers and I didn’t ever use contradiction.