Earlier this week the Ladybug had a bout of RSV, which despite sounding like a new breed of gas-guzzling American car, I have been assured is actually some kind of respiratory infection. The good news is that the Ladybug seems to have shaken the bug off; the bad news is that it apparently landed on me.
As a consequence, I’ve spent this weekend hacking and coughing and oozing snot and feeling just generally cranky. In such cases, the only thing that can make me feel less cranky is to make fun of people who are more cranky that I, and at the top of that list are, of course, mathematical cranks.
It’s fun from time to time to read really bad math, particularly when it’s presented as straight-up correct math. As a current example, I’ve been taking a look at this paper on the arXiv (a popular math and science preprint server), which claims to prove, among other things, that the real numbers are countable.
Here’s the actual title page:
Now, if the title alone didn’t scream Crank! to you in ear-shattering decibels, then certainly the author’s name does. Laurent Germain? Seriously? That’s almost as subtle a pseudonym as, say, Newton Euler-Gauss, or Mathy McCalculustien. Moving on to the abstract…
There’s nothing like an abstract that says “It well known that X has been proven to be true. I will now show X is false.” It reminds me of this joke, the humor of which is clearly lost on the author here. From a strictly pedagogical point of view, if you plan on proving that Condition X is false in spite of the fact that it is known to be true by one of the most famous and studied proofs in history (the Cantor Diagonalization Argument), then there are only three possible things to conclude:
- You screwed up.
- The decades-old, intently-studied, academically accepted proof of the truth of Condition X is logically flawed, in which case the author should point out exactly where the fault in the argument is.
- Condition X is both true and false, meaning that Congratulations! You’ve just proved the whole of mathematics is inconsistent… which is a slightly larger result than the one in the paper.
Which do you think is the most likely?
On to Section 1. Introduction, and more specifically, the second paragraph:
Any mathematical paper with a line like “Infinity can be thought of as an absolute concept and there should not exist several dimensions for the infinite” is crap. What you think should or should not be true is fine for making conjectures, but it has no place in what you end up proving. Mathematical history has actually demonstrated this time and time again. The Greeks thought that for any two like magnitudes there should be a common unit of measure… but the existence of irrational numbers proved otherwise. Geometers thought the parallel postulate should be provable from the other axioms of geometry… but the existence of non-euclidean geometries proved otherwise. Algebraists thought that negative numbers should not have square roots… but the practicality of complex numbers proved otherwise. The mark of a good mathematician is having a good intuitive sense about what should and shouldn’t be true; the mark of a great mathematician is readily accepting the alternative if it is in fact proven true instead. Our author is neither of these.
Essentially, the main construction of the paper is an infinite tree introduced in Section 2, which the author modestly subtitles The Cardinal of the Set of Integers is the same as the Cardinal of the Power set of the Integers. The tree takes the following form: start with a collection of initial nodes, and form each node draw ten branches; then from each of the new nodes so constructed draw ten more branches, and so on and so on. The basic construction is illustrated below.
It is also clearly essential to his argument, because he included it not one, not two, but eight separate times over the course of the proof. His Proposition 2.1 is that the nodes of this infinite tree can be places into a one-to-one correspondence with the set natural numbers {1,2,3,…}, and this is true. For example, the number 991888 is identified with the node we get by starting at node 9, going to sub-node 9, then sub-sub-node 1, and so on, five branches deep. if we follow the branches along the bottom of this diagram 5 levels deep.
The first serious logical flaw is in the next proposition.
Did you catch it? It’s sentence three: “When N is large, counting the number of nodes is the same as counting the number of paths (to infinity) in the tree.” My initial thought to this was the following:
The worst case scenario is that this is a typesetting error, and the sentence should read “When N is large, counting the number of nodes is the same as counting the number of paths (to infinity) in the tree.” This is, of course, utterly stupid, because N, the set of natural numbers, doesn’t “get big.” It doesn’t change, grow, or shrink. It’s a static set of values.
However, granting the benefit of the doubt, the sentence might actually mean “When N is large, counting the number of nodes N levels into the tree is the same as counting the number of paths (to infinity) in the tree.” This is also utterly stupid: just how big does N need to be so that counting more than N things is the same as counting infinitely many of them. Of course, this does present an awesome financial proposition for friends of the author: find out just what this number N is, and then give the author N dollars. Then demand only one-tenth of the money back, but to be paid out in pennies. This means the author owes you 10N pennies, and since 10N > N, by the author’s own admission, he owes you infinitely many pennies. Empty out his bank account and bam! You’re rich!
The rest of Proposition 2.2 is therefore moot, but the problems in the paper don’t end there. Let’s go to the next one.
This one is a bit more subtle. First, you need to believe that the set (0,1)N of all infinite sequences of 0’s and 1’s can be placed into a one-to-one correspondence with the power set P(N), i.e. the set of all subsets of natural numbers. This is easy: simply map a sequence (xn) to the set { n | xn = 1 }.
Next, you need to believe that (0,1)N can be mapped bijectively to the sub-tree consisting of 2 initial nodes with two branches from each node. However, it is not the nodes themselves that one would use to make the bijection; instead, every infinite path through the tree — say the path 0 to 1 to 1 to 0 to 1 to 0 to …. — yields a binary sequence — in this case, (0,1,1,0,1,0,…).
The logical mistake the author makes here is that whereas the set of nodes, which can be thought of as finite path through the tree, that is bijective with N, it is the set of all infinite paths through the two-branch sub-tree that is bijective with (0,1)N. Since the latter set is emphatically not a subset of the former, the rest of the argument is moot.
So we’ve got two critical errors in the first three pages, but like a Ginsu cutlery commercial, Wait! There’s more! The next bit, Section 3: Real numbers, opens with a whopper.
I love the fact that the initial nodes of the tree represent all natural numbers, so obviously there are N+1 of them. Because as was so blatantly asserted in Proposition 2.2, if N is large enough, then N+1 is infinite.
Forgiving the poor choice of words, the basic idea is to repeat the previous table, but with infinitely many more initial nodes, so that a number with the decimal expansion 12.678 would be represented by the node starting at node 12, then going to node 6, then 7, and then 8. What the author asserts next is that every real number can be so described by identifying its decimal expansion with a node on the tree. This is patently false — it’s not even true for rational numbers. The number 1/3 has a non-terminating decimal expansion of 0.333333…, so it cannot be represented by any single node of the tree. In fact, any rational number not of the form n/2p5q has a non-terminating decimal expansion that repeats; and the irrational numbers have decimal expansions that neither terminate nor repeat.
This flaw is implicit in very next paragraph, but it is completely overshadowed by a second, more telling glitch. Can you find it?
Yes, right there on page 4 we have the fact that the number 
At this point, the paper starts to have some real problems…
…But you know what? I’m feeling a bit better now, or at least, a little less cranky.
