Re: abundance of irrationals!)

In article <fb701d3c.0504140901.689de842@xxxxxxxxxxxxxxxxxx>,
mueckenh@xxxxxxxxxxxxxxxxx (W. Mueckenheim) wrote:

> Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message
> news:<ITSnetNOTcom#virgil-A01294.14093213042005@xxxxxxxxxxxxxxxxxxxxxxxx>...
>
> > > Which is irrelevant, because it is characteristic for natural numbers,
> > > that to each n there is an n+1. The n+1th number contains the
> > > antidiagonal formed up to the nth number.
> >
> > But unless it agrees with the entire antidiagonal, not merely a part of
> > it, it is not the same number.
>
> It agrees with the entire antidiagonal, because the entire
> antidiagonal does not agree with the limit 1/9. If you think, however,
> that by some cloudy limit-process the limit 1/9 is the antidiagonal,
> then by the same cloudy limit process my sequence
>
> 0.1;
> 0.10; 0.11;
> 0.001; 0.101; 0.011; 0.111;
> ...
> contains all real numbers of the interval (0,1).

If those are intended to be binary representations, then that list omits
every real in (0,1) whose binary representation requires infinitely many
non-zero digits.
>
> > >
> > > Not the number of mathematicians is important. Most of them know as
> > > little as you did about what Cantor really published.
> >
> > If they have seen any version of the proof and verified it for
> > themselves, whether they saw the original is irrelevant.
>
> The first ones necessarily saw only the wrong version, because there
> was no other. Nevertheless they verified it for themselves.

While there is a flaw in assuming that the diagonal process MUST work
with binary representations, it is fixable in binary , and does not hold
for any base greater than 3. Using bases of 3 or more, the
diagonalization process can be made, and has been made, correct.

Meucken's inability to see this seems to be centered on his inability to
deal with multiple quantification correctly.

Meucken standardly conflates
"for each x there is a y such that ..." with
"there is a y such that for each x ..."

Meuckens inability to distinguish between them marks him as
mathematically incompetent.
> >
>
> > > > And your way fails according to only one non-mathematician.
> > >
> > > Oh no, there are some more people including mathematicians, who think
> > > my way would fail.

While they may have quarrels with actualizing infinities, their quarrels
are different from Meucken's, who hasn't a clue.
> >
> > Who besides yourself thinks the Cantor proof invalid?
>
> You said my way fails to only one non-mathematician. That is
> definitely wrong, as yet.

Who else, mathematician or otherwise, has seem and supported your claims?
> >
> > >
> > > Remember, for 40 years they did believe in a false proof.
> >
> > You only claim the proofs, 1st an 2nd, are false, but have not
> > established either claim.
>
> The first proof did not convince many people.

The first proof is highly technical, and could only be understood by
those with considerable mathematical sophistication. Failing to convince
is not the same as being invalid.

>The second, as Cantor
> published it, used binaries and an exchange of single digits. Think of
> that what you are able to.

I am able to correct any fault you can find in such a proof based on
binary representations by using pairs of digits in the diagonal
construction, effectively converting to base four.

Do you have any URL reference to that 2nd proof showing that it used
binaries? I cannot get Google to point to less than several thousand,
none of which are obviously what is wanted.
>
> Regards, WM
.

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