Re: Ultimate debunking of Cantor's Theory


"Calvin" <crice5@xxxxxxxxxxxxxx> wrote in message news:1184302605.693984.81760@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxx
On Jul 13, 12:27 am, Proginoskes <CCHeck...@xxxxxxxxx> wrote:
On Jul 12, 8:44 am, Calvin <cri...@xxxxxxxxxxxxxx> wrote:
> A variation of that which I subjectively like is making
> it a list of binary expansions instead of decimal. Then
> it is only necessary to 'flip' the diagonal, changing
> all ones to zeros and all zeros to ones.

No, this doesn't work. In fact, it fails spectacularly:

a(1) = .011111...
a(2) = .011111...
a(3) = .011111...
...

Your "new" binary decimal turns out to be .100000..., which is equal
to .01111...; so you don't get a new number after all!

I have no idea what you are talking about.
Your a(1), a(2), and a(3) above do not suggest
a hypothetical list of the binary representations
of the real numbers between 0 and 1, which is what
I was talking about.

I'll take a wild guess at what you mean. Maybe
you are saying that there could be a hypothetical
countable list of the reals between 0 and 1 such
that for the nth element of the list (n>1), the binary
expansion is .0 followed by all 1's out to the
diagonal position, and whatever else beyond the
diagonal position.

But you can't make the list that way, because many
(infinitely many, actually) of the reals between
0 and 1 would be missing from such a list.

Similarly you couldn't make a list of the rational
numbers between 0 and 1 that way. It's absurd.


What Prog said is quite true. He just picked an example which (whilst correct) is probably a little bit too clever.

He hasn't claimed that you can write down a list of all Reals in base 2. He claims that the Cantor diagonal argument cannot be used in base 2 to prove this.

Here is a somewhat clearer example.

Imagine the list is:

a(1) = 0.011111 ...
a(2) = 0.01
a(3) = 0.001
a(4) = 0.0001
..
..

Form the diagonal by flipping bits. You get

cantor diagonal = 0.1000000...

But 0.0111... is the same number 0.1 (unless you also believe that 0.999... <> 1), and so the Cantor diagonal does appear on the list.

Now we all know that the list above doesn't contain every Real, but the Cantor diagonal construction doesn't itself prove this to be true.

(His example was a "bit too clever" because he set a(2) = a(3) = a(4) ...., which is quite valid but obscures his central argument).


.

Relevant Pages

  • Re: Ultimate debunking of Cantors Theory
    ... it a list of binary expansions instead of decimal. ... all ones to zeros and all zeros to ones. ... countable list of the reals between 0 and 1 such ... diagonal position, ...
    (sci.math)
  • Re: Ultimate debunking of Cantors Theory
    ... A countably infinite list of binary expansions of ... you will see that producing even one such list disproves the requirement that the construction always produces the same number. ... except I picked a list with an infinite number of different reals. ...
    (sci.math)
  • Re: Courage?
    ... > This same approach can be done with infinite sets. ... The reals are NOT countable however. ... By then writing the number whose first digit ... Have you heard of leading zeros? ...
    (sci.math)
  • Re: Ultimate debunking of Cantors Theory
    ... argument as used to prove that the reals between 0 and 1 are ... incountallby infinite set with a countable subset deleted is still ... Extend the "width of the list" with appended zeros. ... the diagonal (minus one digit) is an entry of the list. ...
    (sci.math)
  • Re: Non-zero gaps between real numbers
    ... defined for sets does not imply this result (2 from zeros). ... by open intervals whose total length is as small as we please. ... That's a way of saying that there are no gaps in the ... reals. ...
    (sci.math)