In article <email@example.com>, WM <firstname.lastname@example.org> wrote:
> On 12 Dez., 20:33, Alan Smaill <sma...@SPAMinf.ed.ac.uk> wrote: > > > > > And you have forgotten that you claim Cantor uses multiplication > > on reals which are not also cardinals. > > No, that I did not! Here is > my claim: > > AS: Aleph_0 is not a length, nor an area, nor a volume.
That is only one of your many claims. You also claimed that the 2 in 2 times aleph_0 is a real number, not merely a natural, which implies that reals times aleph_0 are defined. > > WM: 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. > > AS: Cantor defined it as a cardinal number; he did not propose any > notion of multiplication of, eg real numbers by transfinite > cardinals. > > WM: You are badly informed.
Not anywhere nearly as badly as WM misinforms himself. > > AS: 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? > > WM: 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)]
2.0 is a real number, but 2 may be only a natural,, or only an integer, or only rational, without also being also a real.
There is a perfectly reasonable point of view in which one starts with the naturals from which one builds integers (or positive rationals) then from one or the other of those builds all rationals, and then from the rationals builds the reals, and so on, with each type separate from all the others.
So that while there is an image of the naturals in the reals, the actual naturals are not reals.
Otherwise when one is talking about the usual reals one would to often have to specify that one means reals which are not also complexes. --