Re: Cantor Confusion


*** T. Winter schrieb:

In article <1162299756.756951.78990@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
>
> *** T. Winter schrieb:
>
> > > For the diagonal number of Cantor's list it is nnot sufficient to come
> > > arbitrarily close to a number which is different from any list number.
> >
> > Sorry, but numbers are fixed and not variable. The sentence "number ...
> > to come arbitrarily close" is nonsense. Numbers do not come arbitrarily
> > close to each other, it is sequences that can come arbitrarily close to
> > each other.
>
> Irrational numbers have no last digit. Therefore, with a sequence of
> digits like the diagonal number is, one can never have a completed
> number but only come as close as possible to any number --- or avoid to
> do so.

The problem is that:
For the diagonal number of Cantor's list it is not sufficient to come
arbitrarily close to a number which is different from any list number
--- or avoid to do so. .


Yes. So what? The sequence comes as close as one wishes to some other
sequence. What is the problem with that?

> > > > Representatives of the equivalence classes that actually are the real
> > > > numbers.
> > >
> > > How many different representatives can be chosen in Cantor's list for
> > > one equivalence class?
> >
> > As many as you want.
>
> Not so. Of course we talk about a fixed base like 10.

Ah. In that case one or two, depending on the number involved. But
the diagonal obviously depends on the actual representative chosen.

So we have no arbitrary choice but , in case of irrational numbers,
exactly one representation. And this representation is *the limit* of
all the sequences of the due equivalence class.

Regards, WM

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