Poor Georg Cantor.

During his life, he suffered from dreadful depression. He was mocked by

his mathematical colleagues, who didn’t understand his work. And after his

death, he’s become the number one target of mathematical crackpots.

As I’ve mentioned before, I get a *lot* of messages either from or

about Cantor cranks. I could easily fill this blog with nothing but

Cantor-crankery. (In fact, I just created a new category for Cantor-crankery.) I generally try to ignore it, except for that rare once-in-a-while that there’s something novel.

A few days ago, via Twitter, a reader sent me a link to a new monstrosity

that was posted to arxiv, called Cantor vs Cantor. It’s novel and amusing. Still wrong,

of course, but wrong in an amusingly silly way. This one, at least, doesn’t *quite*

fall into the usual trap of ignoring Cantor while supposedly refuting him.

You see, 99 times out of 100, Cantor cranks claim to have

some construction that generates a perfect one-to-one mapping between the

natural numbers and the reals, and that therefore, Cantor must have been wrong.

But they never address Cantors proof. Cantors proof shows how, given *any*

purported mapping from the natural numbers to the real, you can construct at example

of a real number which isn’t in the map. By ignoring that, the cranks’ arguments

fail: Cantor’s method still generates a counterexample to their mappings. You

can’t defeat Cantor’s proof without actually addressing it.

Of course, note that I said that he didn’t *quite* fall for the

usual trap. Once you decompose his argument, it does end up with the same problem. But he at least *tries* to address it.

Enough preliminaries. Let’s dive in and see what he did. His abstract

gives about a coherent a description as anything else in the paper, so

we’ll start with that.

Cantor’s diagonal argument makes use of a hypothetical table

T containing all real numbers within the real interval (0,1). That table

can be easily redeﬁned in order to ensure it contains at least all rational

numbers within (0,1). In these conditions, could the rows of T be reordered

so that the resulting diagonal and antidiagonal were rational numbers? In

that case not only the set of real numbers but also, and for the same reason,

the set of rational numbers would be nondenumerable. And then we would

have a contradiction since Cantor also proved the set of rational numbers is

denumerable. Should, therefore, Cantor’s diagonal argument be suspended

until it be proved the impossibility of such a reordering? Is that reordering

possible? This paper address both questions.

To understand this, let’s do a quick review of Cantor’s diagonalization.

Cantor is trying to prove that the set of real numbers is strictly larger than

the set of natural numbers. He uses proof by contradiction: he starts by

supposing that the naturals and the reals have the same size, then shows how

that inevitably leads to a contradiction.

If the set of real numbers is the same size as the set of natural numbers,

then there is a one-to-one mapping f from the natural numbers to the real

numbers in the range (0, 1). So he uses that to lay out a table, where the

first row is f(0), the second row is f(1), etc. In the table, the first column

is the first digit; the second column is the second digit, and so on.

If the mapping is really one-to-one, then every real number must be in the

table. But Cantor shows how you can easily create a new real number which is

*not* in the table. All you do is look at the digit in position (1,1)

in the grid – and change it. Then look at the digit in position (2,2), and

change that. Then the digit in (3,3). And so on: for row N in the table, you

change digit #n. What that procedure does is generate a number which is

different from every number in the table in at least one digit. Therefore it’s

not in the table. That’s a contradiction: we said that every real number had to

be in the table, but we’ve just constructed a real number which isn’t.

What our author is proposing is to take Cantor’s diagonalization,

and do two things to it.

First, he changes it so that it’s a mapping from the natural numbers

to the *rationals* instead of the natural numbers to the reals.

Then, he looks at the diagonal of the and *re-arrange* the rows of it.

He re-arranges the rows of the table until the number in the diagonal is

a rational. Now he’s got a table which contains all of the rationals, and whose

Cantor diagonal is a rational number. So it looks like he’s got a counter-example

for the idea that there’s a one-to-one map between the naturals and the

rationals. If that were the case, then Cantor would be in real trouble: Cantor

also wrote a well-known proof that there’s a one-to-one mapping between the

natural numbers and the rationals. So if our intrepid author is correct, then

either Cantor is wrong about there being *no* mapping between the

naturals and the reals; or he’s wrong about there being a mapping between the

naturals and the rationals; or his entire system of comparing the cardinality

of infinite sets is completely inconsistent.

Looked at naively, it seems sort of compelling: if we can build

a Cantor table that shows that the rationals aren’t countable, then

Cantor is wrong. So what’s wrong with this proof?

Reordering.

Remember: in this proof, we start with a standard Cantor diagonal over the

rationals. That is, we start with an enumeration of rationals, lay it out in a

table, and then read off a number which isn’t in the set of rational numbers.

In other words, we’ve used a Cantor table to produce an irrational number. At

this point, there’s nothing remotely compelling: we *know* that there

are irrational numbers, and all that the construction did at this level is

generate one. This is neither surprising nor particularly interesting, and

it’s certainly no threat to Cantor’s famous proof.

Once he’s got the construction of the irrational, he *re-arranges*

the rows of the table. He tries to re-arrange it so that the number that reads

down the diagonal of the table is a rational. This is exactly the problem: he

*can’t* do that.

Why not? Because the *construction of the re-ordering is invalid*. To

quote the paper:

If it were possible to reorder the rows of T in such a way that a rational antidiagonal

could be defined, then we would have two contradictory results: the set Q of

rational numbers would and would not be denumerable

That is, the re-ordering is absolutely critical to his argument. But

the re-ordering is, itself, self-contradictory.

The argument for the existence of the re-ordering is that

even irrational numbers generally have some probabilistic properties

about their digits. Using those, we can define an initial table

where its counter-example number has a desired set of properties

in the distribution of its digits. (You could use a variety of

properties – but, for example, if the distribution of digits is

uniform, then you could conceptually re-order the rows to produce

0.12345678901234567890…)

So far so good. Now, you re-order the rows, so that the diagonal is

a rational number.

Here’s the problem: you’re constructing a *chosen* rational

number. That is, you *know* what rational number you’re re-ordering the

rows to create. Since it’s a rational, it’s got to be in the table. And since

you *know* what rational it is, you’ve got to know what row in the table

it’s going to be. So go look at that row.

By the definition of the diagonalization, the value of the diagonal *must*

be different from the value of any of the rows by at least one digit. So the

rational number that you’re forming must be different *from itself* by

at least one digit.

Bzzt. No good. The re-ordered rational diagonalization is self-contradictory.

In fact, it’s a classic self-referential foulup.

This is an *obvious* problem, and it’s appalling that the author

of the paper, who is supposedly a *math professor*, couldn’t see it. For all

of the crazy rigamarole he goes through to construct his re-ordering, he never

bothers to look at this simple problem. What kind of mathematician could build

a construction like this and never consider the self-referential case?

I’ll give the author one thing: at least he actually *addressed* Cantor’s

proof. Most authors never bother to do that. Still, he doesn’t really appear to understand

the way that it works – else he’d have have noticed the self-reference problem.

Back at the beginning of the post, I said he *almost* avoids the usual problem of ignoring the diagonalization. The catch is that, as we’ve seen above,

he got it *wrong*, because he didn’t remember to consider the key

property of the diagonalization: that it’s *different* from every row in the

table. By trying to construct a diagonal that *is equal to* a row in the table,

he’s doing something self-contradictory. But he ignores that property – and then when

doing something self-contradictory results in a contradiction, he tries to claim that

it shows that one of history’s most profound and important mathematical results is

wrong.

Bozo.