Virgil
Posts:
9,012
Registered:
1/6/11


Re: fom  01  preface
Posted:
Dec 11, 2012 3:30 AM


In article <2ebb998ad9e9477489e666676e8b7c3b@a6g2000vbh.googlegroups.com>, WM <mueckenh@rz.fhaugsburg.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.fhaugsburg.de> writes: > > >>> On 9 Dez., 23:40, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > > >>>> WM <mueck...@rz.fhaugsburg.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 nonempty > > 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. 

