Re: Review of Mueckenheims book.

On 5 Mrz., 01:22, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1172914305.525476.55...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck....@xxxxxxxxxxxxxxxxx writes:

> > > > But in all this is nonsense. The sequence of digits indeed does not
> > > > converge.
> > >
> > > That means it does not exist as a recognizable entity.
> >
> > Oh. I would think that also non-converging sequences are recognisable
> > entities. Consider the sequence 0 1 0 1 0 1 ..., a pretty recognisable
> > entity, I would think.
>
> Fairly well. We know at least that there appears never a digit 3 in
> it.

So your statement "that means it does not exist as a recognizable entity"
was nonense?


No. To know that there appears never a digit 3 is better than not to
know it but it is not sufficient to recognize the sequence. It is
impossible to determine the digit at position [pi*10^10^100]. And it
is not possible to determine any value as would be possible in case of
0.01010... = 1/99.

> We can never know the digit at position [pi*10^10^100], but we
> know, it will not be 3. With sequences using all digits, there is less
> information, nearly no information at all. Well, we know that there
> will not appear an "a" or so. But is that sufficient for a well-
> defined mathematical entity? With a_n*10^-n the possible error gets
> less than is necessary to mention.

Well, according to Cauchy, yes.

Although he was an engineer, he knew too little about reality. Not his
fault - he lived too early.

>
> Koenig (1905) used this definition to construct a paradox. "Die bisher
> entwickelten Annahmen führen in merkwürdig einfacher Weise zu dem
> Schlosse, daß das Kontinuum nicht wohlgeordnet werden kann."

But *that* is not proven. Moreover, the two quotes are in contradiction
with each other. The first quote states that there must be a definition
that given a number k, gives a real number. The second states that each
definition delivers a different number.

That is the paradox.

> > > > But according to
> > > > this, sqrt(2) is a number.
> > >
> > > Yes.
> >
> > So sqrt(2) is a number after all.
>
> So sqrt(2) is called a number after all.

Above you agred with "this *is* a number". And according to your definition
(trichotomy with other numbers), it *is* a numner.

No. There is only trichotomy with *some* other numbers like 1 or 2 or
1.4142.

Regards, WM

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