Re: abundance of irrationals!)

Virgil <ITSnetNOTcom#virgil@xxxxxxxxxxx> wrote in message news:<ITSnetNOTcom#virgil-90B626.13424406042005@xxxxxxxxxxxxxxxxxxxxxxxx>...

>
> SUM {1/2^k: k in N} is not defined, except as a limit, and that limit
> is equal to 1.

One cannot sum over ALL n?
How, the hell, can one find out whether the series is converging at
all? For this sake one must prove for ALL n > n_0 that ALL the partial
sums S_n satisfy
1 - S_n < eps.
It is incredible! Cauchy's convergence criterion is wrong! Mathematics
has been wrong for 200 years!
No reason to continue to talk about Cantor.

> > > How do you define that sum without using limits?
> >
> > I need not define it. Cantor defined Sum b_k * 10^-k over all k e N,
> > but not including oo.
>
> But Cantor "defined" it only as the limit of the sequence of partial
> sums, since that is the only definition for non-terminating decimals
> extant at that time.
>
> So that if Meuken wants to use Cantor as his model, he must accept the
> same definition Cantor used, which destroys Meucken's case.

Cantor did not bother to define anything like that at all. He
emphasized that for every positive whole number a_nn =/= b_n. That's
all.

> So that if Mueckenheim wants to use Cantor as his model, he must accept the
> same definition Cantor used, which destroys Mueckenheim's case.

Your definition destroys Cantor's case. For a line number n = oo he
would fail to identify a_nn.

Regards, WM
.

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