I’m not the only one who gets mail from crackpots!
A kind reader forwarded me yet another bit of Cantor crackpottery. It never ceases to amaze me how many people virulently object to Cantor, and how many of them just spew out the same, exact, rubbish, somehow thinking that they’re different than all the others who made the same argument.
This one is yet another in the representation scheme. That is, it’s an argument that I can write out all of the real numbers whose decimal forms have one digit after the decimal point; then all of the reals with two digits; then all of them with 3 digits; etc. This will produce an enumeration, therefore, there’s a one-to-one mapping from the naturals to the reals. Presto, Cantor goes out the window.
As usual, the crank starts off with a bit of pomposity:
My mathematic researshes lead me to resolve the continuum theory of Cantor, subject of controversy since a long time.
This mail is made to inform the mathematical community from this work, and share the conclusions.
You will find in attachment extracts from my book “Théorie critique fondamentale des ensembles de Cantor”,
Inviting you to contact me,
Member of the American mathematical society
Membre de la société mathématique de France
Member of the Bulletin of symbolic logic
Director of éditions européennes
As a quick aside, I love how he signs he email “Member of the AMS”, as if that were something meaningful. The AMS is a great organization – but anyone can be a member. All you need to do is fill out a form, and write them a check. It’s not something that anyone sane or reasonable brags about, because it doesn’t mean anything.
Anyway, let’s move on. Here’s the entirety of his proof. I’ve reproduced the formatting as well as I could; the original document sent to me was a PDF, so the tables don’t cut-and-paste.
The well-order on the set of real numbers result from this remark that it is possible to build, after the comma, a set where each subset has the same number of ordered elements (as is ordered the subset 2 : 10 …13 … 99).
Each successive integer is able to be followed after the comma (in french the real numbers have one comma after the integer) by an increasing number of figures.
0,0 0,10 0,100 0,1 0,11 0,101 0,2 0,12 0,102 … … … 0,9 0,99 0,999
It is the same thing for each successive interger before the comma.
1 2 3
So it is the 2 infinite of real number.
For this we use the binary notation.
But Cantor and his disciples never obtained this simple result.
After that, the theory displays that the infinity is the asymptote of the two branches of the hyperbole thanks to an introduction of trigonometry notions.
The successive numbers which are on cotg (as 1/2, 1/3, 1/4, 1/5) never attain 0 because it would be necessary to write instead (1/2, 1/3, 1/4, 1/4 ).
The 0 of the cotg is also the origin of the asymptote, that is to say infinite.
The beginning is, pretty much, a typical example of the representational crankery. It’s roughly a restatement of, for example, John Gabriel and his decimal trees. The problem with it is simple: this kind of enumeration will enumerate all of the real numbers with finite length representations. Which means that the total set of values enumerated by this won’t even include all of the rational numbers, much less all of the real numbers.
(As an interesting aside: you can see a beautiful example of what Mr. Collot missed by looking at Conway’s introduction to the surreal numbers, On Numbers and Games, which I wrote about here. He specifically deals with this problem in terms of “birthdays” and the requirement to include numbers who have an infinite birthday, and thus an infinite representation in the surreal numbers.)
After the enumeration stuff, he really goes off the rails. I have no idea what that asymptote nonsense is supposed to mean. I think part of the problem is that mr. Collot isn’t very good at english, but the larger part of it is that he’s an incoherent crackpot.