Re: Review of Mueckenheims book.

In article <1173165366.640448.190440@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
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.

Sorry, but you stated that it did not exist *because* the sequence of
digits does not converge. Was that nonsense or not?

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.

End that was specifically *not* what was under discussion.

> 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.

Mathematics is *not* engineering.

> Koenig (1905) used this definition to construct a paradox. "Die bisher
> entwickelten Annahmen f=FChren in merkw=FCrdig einfacher Weise zu dem
> Schlosse, da=DF 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.

No, that is not the paradox, that is due to a faulty definition.

> > > > 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.

You agreed that it *is* a number, see your statement above: "Yes.". But
with what number is there no trichotomy?
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
.