May 05 2013 Published by under Bad Math, Cantor Crankery

A fun game to play with cranks is: how long does it take for the crank to contradict themselves?

When you're looking at a good example of crankery, it's full of errors. But for this game, it's not enough to just find an error. What we want is for them to say something so wrong that one sentence just totally tears them down and demonstrates that what they're doing makes no sense.

"The color of a clear sky is green" is, most of the time, wrong. If a crank makes some kind of argument based on the alleged fact that the color of a clear daytime sky is green, the argument is wrong. But as a statement, it's not nonsensical. It' just wrong.

On th other hand, "The color of a clear sky is steak frite with bernaise sauce and a nice side of roasted asparagus", well... it's not even wrong. It's just nonsense.

Today's crank is a great example of this. If, that is, it's legit. I'm not sure that this guy is serious. I think this might be someone playing games, pretending to be a crank. But even if it is, it's still fun.

About a week ago, I got en mail titled "I am a Cantor crank" from a guy named Chris Cuellar. The contents were:


Haha, ok, not exactly. But you really seem to be interested in this stuff. And so am I. But I think I've nailed Cantor for good this time. Not only have I come up with algorithms to count some of these "uncountable" things, but I have also addressed the proofs directly. The diagonalization argument ends up failing spectacularly, and I believe I have a good explanation for why the whole thing ends up being invalid in the first place.

And then I also get to the power set of natural numbers... I really hope my arguments can be followed. The thing I have to emphasize is that I am working on a different system that does NOT roll up cardinality and countability into one thing! As it will turn out, rational numbers are bigger than integers, integers are bigger than natural numbers... but they are ALL countable, nonetheless!

Anyway, I had started a little blog of my own a while ago on these subjects. The first post is here:

Have fun... BWAHAHAHA

So. We've got one paragraph of intro. And then everything crashes and burns in an instant.

"Rational numbers are bigger than integers, integers are bigger than natural numbers, but they are all countable". This is self-evident rubbish. The definition of "countable" say that an infinite set I is countable if, and only if, you can create a one-to-one mapping between the members of I and the natural numbers. The definition of cardinality says that if you can create a one-to-one mapping between two sets, the sets are the same size.

When Mr. Cuellar says that the set of rational numbers is bigger that the set of natural numbers, but that they are still countable... he's saying that there is not a one-to-one mapping between the two sets, but that there is a one-to-one mapping between the two sets.

Look - you don't get to redefine terms, and then pretend that your redefined terms mean the same thing as the original terms.

If you claim to be refuting Cantor's proof that the cardinality of the real numbers is bigger than the cardinality of the natural numbers, then you have to use Cantor's definition of cardinality.

You can change the definition of the size of a set - or, more precisely, you can propose an alternative metric for how to compare the sizes of sets. But any conclusions that you draw about your new metric are conclusions about your new metric - they're not conclusions about Cantor's cardinality. You can define a new notion of set size in which all infinite sets are the same size. It's entirely possible to do that, and to do that in a consistent way. But it will say nothing about Cantor's cardinality. Cantor's proof will still work.

What my correspondant is doing is, basically, what I did above in saying that the color of the sky is steak frites. I'm using terms in a completely inconsistent meaningless way. Steak frites with bernaise sauce isn't a color. And what Mr. Cuellar does is similar: he's using the word "cardinality", but whatever he means by it, it's not what Cantor meant, and it's not what Cantor's proof meant. You can draw whatever conclusions you want from your new definition, but it has no bearing on whether or not Cantor is correct. I don't even need to visit his site: he's demonstrated, in record time, that he has no idea what he's doing.

6 responses so far

  • Richard says:

    If you were to frappe steak frites with bernaise sauce in a blender, what color would it be?

  • theshortearedowl says:


  • Jonathan D says:

    Having looked at his blog, I'm inclined to agree that it probably isn't genuine.

  • Joel says:

    From reading his blog, he did seem to genuinely believe what he was saying. Now, from what I can tell, his system fails because it doesn't take into account that whereas integers each have a finite length, whereas a real has an actually infinite length. A common enough flaw in Cantor counter-arguments.

    He then argues against Cantor's diagonalisation argument by thinking that the number produced by Cantor's diagonlisation process is some sort of constant, which has strange and inconsistent properties if you should reorder the mapping. However, diagonlisation doesn't produce a constant, it's just a system for generating numbers which are not present in the mapping - of which there are, conveniently, infinitely many, for any attempted mapping.

    If you take a mapping, using diagonlisation to find a number that is not in the mapping, reorder the mapping, and then using diagonalisation to generate a new number, it would select a different real that also does not appear in the mapping. That's real, the original real is still real, and neither occur in the reordered mapping.

  • Jonathan D says:

    Oh yes, he repeatedly writes as though the natural numbers include some sort of numbers that are 'infinite', rather than simply arbitrarily large. As he talks about them, they probably are of the same cardinality of the reals - the problem just comes when you try to square that with what anyone else means by natural numbers or a sequence, or, as Marks says, cardinality and countability.

    But it's one thing to "plead with us to understand" this the "theoretical possibility" of his version of a sequence which reaches all these repeating decimals and irrational numbers, and it's another to conveniently forget all these reals when trying to show us how diagonalisation falls down. Diagonalisation can be confusing at first (I can understand the hangup treating hte number as constant), but forgetting the point you've been trying to convince us of is trying a bit too hard, either deliberately or otherwise.

    Then again, it's his next post that really doesn't seem sincere.

  • JamesT says:

    The beauty of a crank is that it is difficult to tell if they are serious or not. It seems that for any crank we think is not serious because of their funny contradictory statements, we can find another crank who makes more of them and does seem to be serious.

    Anyway, I just discovered your blog and love it, thanks.

    Oh I recently found this ancient article, which is kind of a tangent to this topic:
    "A Measure for Crackpots" by Fred Gruenbergers in 1962: