Re: Ultimate debunking of Cantor's Theory

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.

.

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: 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)
  • Re: rational or irrational?
    ... >a sequence of base 10 digits) does not exist in our universe. ... The "sequence" described claims that there is "an infinite ... amount of zeros", ... >My point here is infinite digits is not void in subjects of reals, ...
    (sci.math)