Re: An uncountable countable set


*** T. Winter schrieb:

In article <1152736417.397578.308650@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
>
> *** T. Winter schrieb:
>
> > > > > I proved in my special list even that the diagonal number is a
> > > > > rational.
> > > >
> > > > I wonder whether it was a proof or just some handwaving.
> > >
> > > 0.0
> > > 0.1
> > > 0.11
> > > 0.111
> > > ...
> > > replace 0 by 1.
> >
> > As far as I see the diagonal starts with 1.000... Am I right?
>
> I use only the digits behind the point: So the diagonal is 0.111... =
> 1/9.

So you imply additional 0's after your notation. I was not sure.

> If there is no other outcome possible, I don't need a further
> definition. 0.999... = 1 follows from the definition of (+,-,*,/) in
> the real numbers.

Pray tell me how. Somewhere else you stated that the representation
0.111... did not exist. What are you arguing? Either the representation
0.111... does exist or not. And if it does exist the definitions of the
mathematical operations are not sufficient to give a meaning to it.

Anyhow, how do you show that 1.000... - 0.999... = 0 with the definitions
you are using?

In "usual" mathematics we have 10 * 0.999... = 9.999... and from that
we get easily 0.999... = 1.

I argue that 0.111... does not exist. When I will have succeded, also
0.999... will be abolished. But that has not yet been generally
accepted. OK?


Regards, WM

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