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Topic: fom - 01 - preface
Replies: 18   Last Post: Dec 12, 2012 3:34 PM

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Posts: 18,076
Registered: 1/29/05
Re: fom - 01 - preface
Posted: Dec 12, 2012 12:22 PM
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On 12 Dez., 17:00, fom <> wrote:

> >>>> Find a different criticism of Alan's remarks
> >>>> if you must.  This one is incorrect.

> >>> So you disagree that 2 is a real number?

Obviously you do not.
> >> Since you like quoting the Grundlagen, try
> >> transcribing long detailed passages from
> >> section 9

> > I have written read an written everything Cantor wrote.
> >> where Cantor rejects definitions
> >> that conflate logical priority as you have
> >> been doing.

> > I have not been doing so. At that time there was no difference between
> > reals, integers and cardinals (because Cantor did not suspect that
> > there would apperar a contradiction). He just had switched from oo to
> > omega. No alpphs in sight.

> >> That is where he calls his construction
> >> "... a fundamental sequence and correlate
> >> it with a number b, TO BE DEFINED THROUGH
> >> IT,..."

> > And those numbers are multiplied by real numbers.
> wrong
> the fundamental sequences ARE the real numbers

> > [text unrelated to the topic deleted]
> Quite wrong.

> That was the text that explained how the well-ordered
> set that is referenced in 2*omega is
> not described as a fundamental sequence.

That is not under discussion here. We discussed the question whether
Cantor multiplied his transfinite numbers with real numbers. And that
he did not only, but called the new realm of numbers also "ganze
Zahlen" whole numbers or integers, and he extended these transfinite
integers by addition of fractions to what?
> Cite the sections from which
> you are quoting.  And quote significantly
> lengthy passages so that the text is
> in context.-

All my quotings were from the 1883 Grundlagen. You seem to be in
possession of this paper. I have not the English translation.
Therefore it would be boring for most treaders to quote extended
paragraphs here.

Only so much:
G. Cantor: Grundlagen einer allgemeinen Mannigfaltigkeitslehre
(Leipzig 1883).
§ 1, first sentence: Die bisherige Darstellung meiner Untersuchungen
in der Mannigfaltigkeitslehre ist an einen Punkt gelangt, wo ihre
Fortführung von einer Erweiterung des realen ganzen Zahlbegriffs über
die bisherigen Grenzen hinaus abhängig wird, und zwar fällt diese
Erweiterung in eine Richtung, in welcher sie meines Wissens bisher
noch von niemandem gesucht worden ist. [...] Denn es handelt sich um
eine Erweiterung resp. Fortsetzung der realen ganzen Zahlenreihe über
das Unendliche hinaus; so gewagt dies auch scheinen möchte,

He says that he extends the notion of real whole number.

But of course he does not do so. He was in error. "We" know it better
today. And why must "we" know it better than Cantor did? Because he
did not yet know that his theory is self-contradictory when real
numbers are called real numbers.

§ 4: Die erweiterte ganze Zahlenreihe kann, wenn es die Zwecke
fordern, ohne weiteres zu einer kontinuierlichen Zahlenmenge
vervollständigt werden, indem man zu jeder ganzen Zahl alpha alle
reellen Zahlen x, die größer als Null und kleiner als Eins sind,

What a shame! My recommendation: Don't take notice. Simply continue to
argue as if he had never said that. As ususal. Otherwise you could
feel obliged to apologize. That is what civil people do when they are

Regards, WM

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