Re: Review of Mueckenheims book.

In article <1172914305.525476.55260@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueckenh@xxxxxxxxxxxxxxxxx writes:
On 3 Mrz., 02:54, "*** T. Winter" <***.Win...@xxxxxx> wrote:
In article <1172825991.885766.236...@xxxxxxxxxxxxxxxxxxxxxxxxxxxx> mueck...@xxxxxxxxxxxxxxxxx writes:
> On 2 Mrz., 02:22, "*** T. Winter" <***.Win...@xxxxxx> wrote:
> > In article <1172678208.052350.316...@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?

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.

> > But as Cantor's original proof was not
> > about real numbers, there is no problem.
>
> Then it was about undefined non-converging sequences which do not
> belong to the stuff which can be dealt with in mathematics.

Oh. Again something stated as judg(e)ment, without any justification.

See above. But even without environmental restrictions a non
converging sequence is undefined. Why does mathematicians prefer to
use converging series? They simply cannot handle non-converging
series.

They can. Why do you think they can not? A series like the 'w', 'm'
series if just a mapping f: N -> {'w', 'm'}.

> Do you have an idea what the meaning of "all" is? It is meaningless to
> use the quantifier "all" in connection with a statement, if it is
> impossible to verify that all elements satisfy this statement.

That is the finitistic approach (or MatheRealism). How do you define
limits in MatheRealism?

I do not change mathematics! It remains at is. I only hope to raise
the consciousness that not everything can be done to any epsilon. For
all practical purposes FAPP epsilon < 10^-50 is sufficient and in
most cases achievable.

In that case, show a *proof* of the triangle inequality.

> I defined it by "finite definition"

Non-standard terminology, as I stated. But you really should look at
"computable number" where such things have been rigorously defined.
Because (as you show in your book) the term "finite definition" alone
leads to paradoxes:
the smallest number that can not be defined with sixty-six symbols
What is the number?

Here is the original definition by Koenig:
"Ein Element des Kontinuums soll ,,endlich definiert" heißen, wenn wir
mit Hilfe einer zur Fixierung unseres wissenschaftlichen Denkens
geeigneten Sprache in endlicher Zeit ein Verfahren (Gesetz) angeben
können, das jenes Element des Kontinuums von jedem anderen begrifflich
sondert, oder - anders ausgedrückt - für ein beliebig gewähltes k die
Existenz einer und nur einer zugehörigen Zahl a ergibt."

So your "finite definition" was *not* sufficient, and you need a more
rigorous definition. However, if I read that statement correctly (the
last part), he wants to have a finitely defined mapping from N to R.
And so his set of "finitely defined" numbers is a "finitely defined"
mapping from N to R. But as there can be multiple such mappings, the
definition is not sufficient.

> > Using non-standard terminology.
>
> Using Koenig' terminology: "Eine solche endliche Definition muss
> nämlich durch eine endliche Zahl von Buchstaben und
> Interpunktionszeichen, die selbst nur in bestimmter, endlicher Zahl
> vorhanden sind, vollständig gegeben sein. Man kann weiter jene
> verschiedenen endlichen Definitionen wieder so anordnen, dass jeder
> beliebigen Definition eine und nur eine bestimmte positive ganze Zahl
> als (endliche) Ordnungszahl entspricht."

Yes, using non-standard terminolgy. I will grant that König possibly
did not understand that his definition would lead to paradoxes.

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.

> > 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.
--
*** t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131
home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~***/
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