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

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Virgil

Posts: 7,014
Registered: 1/6/11
Re: fom - 01 - preface
Posted: Dec 10, 2012 3:53 PM
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In article
<e9c17538-bdb5-443e-a1bc-372222b58db0@h5g2000vbb.googlegroups.com>,
WM <mueckenh@rz.fh-augsburg.de> 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)]


In order to define the product of a real number times a transfinite, the
definition must hold for all reals and all transfinites.

In order to define the product of a natural and a transfinite the
definition must hold for all naturals and all transfinites.

2 and 3 are also complex numbers and quaternions, so that, according to
WM, Cantor defined both complex times transfinite and quaternion times
transfinite multiplications.
--





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