Re: Cantor's diagonal proof wrong?
From: Chairman of the David Hilbert Appreciation Society (mathgeekxxiiii_at_hotmail.com)
Date: 11/14/04
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Date: Sun, 14 Nov 2004 13:47:34 -0500
Curt Welch wrote:
> Here's something all of you should have some fun with.
[...]
A quick note on how my comments relate to your notation:
You're refering to the set {0, 1, 2, 3, ...} as the integers,
but this set is known as the naturals and is denoted by N. The
integers would include negative values as well
{..., -2, -1, 0, 1, 2, ...}, and is denoted by Z.
I'll refer to the set {0, 1, 2, ...} as N. I'll also refer
to the rationals as Q. Keep these changes in mind to minimize
confusion.
[...]
> I claim that there is only one type of infinity. That there are just as
> many integers as there real numbers. (or more accurately, that the concept
> of the size of an infinite set is a contradiction in itself).
>
> So, let me create a mapping. I'll start with the mapping from the integers
> to the reals in the range 0 to 0.99999....
>
> I R
>
> 0 0.0000...
> 1 0.1000...
> 2 0.2000...
>
> 10 0.0100...
>
> 123 0.3210...
You'll notice that each natural on the left corresponds to a finite
length decimal representation on the right however not all real
numbers can be represented by this method.
More on this below.
[...]
> Let's look at this mapping with Cantor's diagonal proof. We construct a
> real number by picking digits from the diagonal which is different from
> each row in the table. Well, as it happens, the diagonal in this mapping
> is all zeros[...]
Exactly. Because every number on the right side of your list
is a rational. You're putting N into one-to-one correspondence
with a subset of Q, not R.
[...]
> As another example, let me show that the number of integers are also
> greater than the number of integers, using the logic of Cantor's proof.
>
> Lets create a table of integers like this:
>
> ...000000
> ...000001
> ...000002
>
> ...000010
>
> ...000123
>
> It's just a normal list of integers, but instead of following the normal
> convention of leaving off the leading zeros (which we all know are implied
> even if we don't write them) I include them in that table.
>
> So lets use Cantor's logic on this table and see if we can construct a
> number which is not in the table. We take the numbers from the diagonal,
> and construct the number ...111111 just like we did above.
How many digits does this diagonal string have?
It has infinitely many digits.
We agree that it's not on the list, however it's not a natural
number and so we can't conclude that it represents a natural
number that _isn't on the list_.
The key difference between Cantor's diagonal proof with the reals
and your attempted use of that proof strategy with integers is
that the integers have finite length representations and the
reals don't.
When we diagonalize we get an infinite length string...
Think about this please, don't just react.
[...]
> Much other important work, such as Gödel's, also fell prey to this same
> mistake.
Let's just deal with Cantor first.
> Oh, and if you want a mapping from the integers to all the reals, here's
> one:
>
> 0 0.0
> 1 0.1
> 10 1.0
> 123 2.31
>
> i.e., you take the integer and number the digits like this:
>
> ... D4 D3 D2 D1 D0
>
> And you construct the real as: ... D3 D1 . D0 D2 D4 ...
Your mistake is common. The problem that most people seem to
have with Cantor's well known diagonal proof is that they
don't know what the real numbers are. I'm not joking about this.
This is especially so for people who aren't mathematicians
yet have a technical undergrad background which gives them
exposure to Cantor's diagonal proof, without a proper development
of the real numbers. Following graduation the student has great
confidence in his technical abilities, but doesn't actually know
what the real numbers are.
His exposure to the properties of real numbers is based on
informally taught subjects like Algebra, Trig, and Calc where
every variable and every answer to every problem is shown on
the tiny calculator screen as a finite length decimal expansion.
It's my belief that the repetition of a student seeing "any real
number" expressed as finite string on an LCD display combined
with his informal exposure to the subject results in his developing
a powerfully felt conviction that any/all real numbers are finite
length decimal expansions.
Finite length decimal expansions are a subset of rational numbers,
they are countable. Cantor essentially proved this in 1873.
The fact that Cantor proved a result which you intuitively expect
should give you greater confidence in his work.
[...]
> Has any one else put forth this same argument (or others) that Cantor's
> proof is invalid?
Yes. This happens often in sci.math. People mistakenly think
that the set of real numbers is the set of all finite decimal
representations. By showing a one-to-one correspondence between
N and all finite decimal representations you're rediscovering
a 125 year old result.
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