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

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Virgil

Posts: 7,011
Registered: 1/6/11
Re: fom - 01 - preface
Posted: Dec 11, 2012 3:30 AM
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In article
<2ebb998a-d9e9-4774-89e6-66676e8b7c3b@a6g2000vbh.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> wrote:

> On 10 Dez., 21:03, fom <fomJ...@nyms.net> wrote:
> > On 12/10/2012 11:57 AM, WM wrote:
> >
> >
> >
> >
> >

> > > 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)]

> >
> > Yes. He did. But, Cantor's notion of a real
> > number was clearly found in the completion of a
> > Cauchy space.

>
> That is completely irrelevant for the result.


Either than or the Dedekind cut model was necessary back then as they
were the only satisfactory models then.
>
> >  He found that more appealing
> > than Dedekind cuts.  This is evident since
> > his topological result of nested non-empty
> > closed sets in a complete space is closely
> > related.
> >
> > There are ordinal numbers in set theory given
> > the names of natural numbers.

>
> Only those which are finite.

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

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


Whether it is a real number depends on the context, and there are a lot
of contexts in which it is not.

There are a lot of rings or multiplicative groups with 1 as thei
multiplicative identity which do not contain any real numbers at all.
--





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