Re: An uncountable countable set


*** T. Winter schrieb:

> > > And what is the consequence of this?
> >
> > That the calculation of the digits can be done in parallel?
>
> It is done in zero time. It is determined from the beginning. Why
> should something "be done"?

It was you who remarked on the calculations...

The result of a calculation does not depend on the time you need to
obtain it.

> > > > It is Cauchy's theorem that proves that the limit exists, using the
> > > > epsilon argument.
> > >
> > > But Cantor's arguing is that without epsilon.
> >
> > I do not know Cantor's argument exactly. I think that he implicitly
> > uses that result. In my formulation it was abundantly clear that I did
> > use it.
>
> But he needs to consider every digit with equal weight. That is not a
> limit process.

What did he *mean* when he wrote "with equal weight"?

He did not write that. I used this description to express that all
digits must be distinguishable from the exchanged digits and that the
resulting number must be distinguishable from the number with one digit
left unexchanged, be it the first or the last one (which can be
recognized).

By the way, Virgil's arguing leads to the "paradox": There is no last
number and not that one next to the last and so on. Which one is the
first one that does not exist?
> I define limit by: *Using all finite natural numbers* just as like as
> Cantor does.

No. You do *not* and he does *not*. In the Cantor diagonal the number
obtained is a real number using limits (in the Cauchy sense) with the
definition of real number by many others (that can be proven to all be
equivalent with each other). The limit used by Cantor is precisely the
epsilon argument, together with majoration and minoration on the ordered
set of rational numbers. By Cauchy any sequence of decimal digits has a
limit, but it is not certain whether that limit is in the defined set of
numbers. By Cantor, Dedekind, Weierstrass and a host of others (using
various formulations), such a limit (when starting with rational numbers)
is defined as a real number. That is the place where Cantor uses the
limit (although he may not have expressed it as such), i.e. showing that
the resulting number is a real.

Cantor does not at all use real numbers but merely infinite sequences
with only two different symbols w and m (which might be interpreted as
binary representations but were not). But that there is no satisfactory
limit consideration becomes clear from the following: We know that
0.999... = 1.000... This leads to the result that a change of 1 in the
limit where the digit number goes to oo does not have the effect which
would be required in order to distinguish the diagonal number from the
list numbers.

But in order to concern all lines of the list, n has to go to oo
(though never reach it) just as in case of my example.

Regards, WM

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