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

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mueckenh@rz.fh-augsburg.de

Posts: 16,056
Registered: 1/29/05
Re: fom - 01 - preface
Posted: Dec 10, 2012 12:57 PM
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On 10 Dez., 18:24, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> WM <mueck...@rz.fh-augsburg.de> writes:
> > On 9 Dez., 23:40, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote:
> >> WM <mueck...@rz.fh-augsburg.de> writes:
> >> > On 8 Dez., 23:37, Virgil <vir...@ligriv.com> wrote:
> >> >> Aleph_0 is not a length, nor an area, nor a volume.
>
> >> > If it was a whole number or integer, as Cantor insisted, then it could
> >> > be used to define a length or an area or a volume etc.

>
> >> Cantor defined it as a cardinal number;
> >> he did not propose any notion of multiplication of,
> >> eg real numbers by transfinite cardinals.

>
> > You are badly informed.
>
> Then please inform me;
> did Cantor consider 3.14159...  to be a cardinal number?
> In which of Cantor's number classes does 3.14159... fall?


First you said something else, namely: "he did not propose any notion
of multiplication of, eg real numbers by transfinite cardinals". This
claim is wrong because 2, 3, .. are real numbers. Cantor defined
2*omega, 3*omega, ...
[Grundlagen einer allgemeinen Mannigfaltigkeitslehre (Leipzig 1883)]

But in § 4 of the same paper he even wrote: 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, hinzufügt.

Of course also 3.14159... can be added to the extended (transfinite)
integer alpha.

Regards, WM





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