Re: Review of Mueckenheims book.

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:
>
> > Private Microsoft encoding. I assume a horizontal arrow.
>
> Excuse me, I will return to "-->".

> > > Two series which differ by lim {n->oo) 10^-n are he same. Compare
> > > 1.000... and 0.999.... Two sequences which differ by 1 at the same
> > > position are different or must be considered different in order to
> > > save Cantor's proof.
> >
> > 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. 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.

> > The sequence of digits of a real number in decimal converges
> > if and only if the digits remain ultimately the same. That is, only for
> > those rationals where the denumerator (in simplest form) is either the
> > product of a number of factors 2 and a number of factors 5, or such a
> > product multiplied by either 3 or 9. The number is indeed defined by that
> > infinite sum. Yes, and in that proof, sequences like 1.000... and
> > 0.999... must be considered different.
>
> That is decisive!
>
> > 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.

> > > > If you refuse to use standard mathematical terminology... The digits
> > > > can be constructed, all of them.
> > >
> > > Including the last one? You certainly mean "every" digit.
> >
> > As there is no last digit, I have no idea about what you are meaning here.
>
> 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.

> This
> cannot be proven other than by looking at every one and being finished
> at some time.

So limits can not be defined in MatheRealism?

> > > > That does *not* make the number
> > > > constructable, and also not even computable.
> > >
> > > A number every digit of which can be constructed I call constructible.
> >
> > Using non-standard terminology.
>
> 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."

> > > A definition which cannot be given by a finite number of words I call
> > > a non-definition.
> >
> > 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."

> > > A number which cannot be compared by trichotomy with all other numbers
> > > I call a non-number.
> >
> > So you are really using the term "complex non-number"?

No, I use "complex 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.

Regards, WM

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