For every natural number N, there's a Cantor Crank C(n)

Nov 10 2012 Published by under Bad Logic, Bad Math, Cantor Crankery

More crankery? of course! What kind? What else? Cantor crankery!

It's amazing that so many people are so obsessed with Cantor. Cantor just gets under peoples' skin, because it feels wrong. How can there be more than one infinity? How can it possibly make sense?

As usual in math, it all comes down to the axioms. In most math, we're working from a form of set theory - and the result of the axioms of set theory are quite clear: the way that we define numbers, the way that we define sizes, this is the way it is.

Today's crackpot doesn't understand this. But interestingly, the focus of his problem with Cantor isn't the diagonalization. He thinks Cantor went wrong way before that: Cantor showed that the set of even natural numbers and the set of all natural numbers are the same size!

Unfortunately, his original piece is written in Portuguese, and I don't speak Portuguese, so I'm going from a translation, here.

The Brazilian philosopher Olavo de Carvalho has written a philosophical “refutation” of Cantor’s theorem in his book “O Jardim das Aflições” (“The Garden of Afflictions”). Since the book has only been published in Portuguese, I’m translating the main points here. The enunciation of his thesis is:

Georg Cantor believed to have been able to refute Euclid’s fifth common notion (that the whole is greater than its parts). To achieve this, he uses the argument that the set of even numbers can be arranged in biunivocal correspondence with the set of integers, so that both sets would have the same number of elements and, thus, the part would be equal to the whole.

And his main arguments are:

It is true that if we represent the integers each by a different sign (or figure), we will have a (infinite) set of signs; and if, in that set, we wish to highlight with special signs, the numbers that represent evens, then we will have a “second” set that will be part of the first; and, being infinite, both sets will have the same number of elements, confirming Cantor’s argument. But he is confusing numbers with their mere signs, making an unjustifiable abstraction of mathematical properties that define and differentiate the numbers from each other.

The series of even numbers is composed of evens only because it is counted in twos, i.e., skipping one unit every two numbers; if that series were not counted this way, the numbers would not be considered even. It is hopeless here to appeal to the artifice of saying that Cantor is just referring to the “set” and not to the “ordered series”; for the set of even numbers would not be comprised of evens if its elements could not be ordered in twos in an increasing series that progresses by increments of 2, never of 1; and no number would be considered even if it could be freely swapped in the series of integeres.

He makes two arguments, but they both ultimately come down to: "Cantor contradicts Euclid, and his argument just can't possibly make sense, so it must be wrong".

The problem here is: Euclid, in "The Elements", wrote severaldifferent collections of axioms as a part of his axioms. One of them was the following five rules:

  1. Things which are equal to the same thing are also equal to one another.
  2. If equals be added to equals, the wholes are equal.
  3. If equals be subtracted from equals, the remainders are equal.
  4. Things which coincide with one another are equal to one another.
  5. The whole is greater that the part.

The problem that our subject has is that Euclid's axiom isn't an axiom of mathematics. Euclid proposed it, but it doesn't work in number theory as we formulate it. When we do math, the axioms that we start with do not include this axiom of Euclid.

In fact, Euclid's axioms aren't what modern math considers axioms at all. These aren't really primitive ground statements. Most of them are statements that are provable from the actual axioms of math. For example, the second and third axioms are provable using the axioms of Peano arithmetic. The fourth one doesn't appear to be a statement about numbers at all; it's a statement about geometry. And in modern terms, the fifth one is either a statement about geometry, or a statement about measure theory.

The first argument is based on some strange notion of signs distinct from numbers. I can't help but wonder if this is an error in translation, because the argument is so ridiculously shallow. Basically, it concedes that Cantor is right if we're considering the representations of numbers, but then goes on to draw a distinction between representations ("signs") and the numbers themselves, and argues that for the numbers, the argument doesn't work. That's the beginning of an interesting argument: numbers and the representations of numbers are different things. It's definitely possible to make profound mistakes by confusing the two. You can prove things about representations of numbers that aren't true about the numbers themselves. Only he doesn't actually bother to make an argument beyond simply asserting that Cantor's proof only works for the representations.

That's particularly silly because Cantor's proof that the even naturals and the naturals have the same cardinality doesn't talk about representation at all. It shows that there's a 1 to 1 mapping between the even naturals and the naturals. Period. No "signs", no representations.

The second argument is, if anything, even worse. It's almost the rhetorical equivalent of sticking his fingers in his ears and shouting "la la la la la". Basically - he says that when you're producing the set of even naturals, you're skipping things. And if you're skipping things, those things can't possible be in the set that doesn't include the skipped things. And if there are things that got skipped and left out, well that means that it's ridiculous to say that the set that included the left out stuff is the same size as the set that omitted the left out stuff, because, well, stuff got left out!!!.

Here's the point. Math isn't about intuition. The properties of infinitely large sets don't make intuitive sense. That doesn't mean that they're wrong. Things in math are about formal reasoning: starting with a valid inference system and a set of axioms, and then using the inference to reason. If we look at set theory, we use the axioms of ZFC. And using the axioms of ZFC, we define the size (or, technically, the cardinality) of sets. Using that definition, two sets have the same cardinality if and only if there is a one-to-one mapping between the elements of the two sets. If there is, then they're the same size. Period. End of discussion. That's what the math says.

Cantor showed, quite simply, that there is such a mapping:

\[{ (i rightarrow itimes 2) | i in N }\]

There it is. It exists. It's simple. It works, by the axioms of Peano arithmetic and the axiom of comprehension from ZFC. It doesn't matter whether it fits your notion of "the whole is greater than the part". The entire proof is that set comprehension. It exists. Therefore the two sets have the same size.

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