Walter Felscher walter.felscher at
Tue Mar 23 10:47:01 EST 1999

On Mon, 22 Mar 1999, Alexander Zenkin wrote among many other things:

>                                         But he <Cantor> disregards all
> these contradictions and absurdities, and boldly goes further: applying
> the common operation "+1", defined for finite integers, to the
> transfinite "NUMBER" W, he obtains, by means of, supposedly, his "FIRST
> generating principle" (i.e., by means of the common Peano's axiom
> defined for the finite natural numbers: "IF n THEN n+1"), further
> numbers:
> W, W+1, W+2 , ... , W+n , ... , 2W , 
 , 3W , 
 , nW , 
 , W^2 , 
, W^3
> , 
 , W^W , 
 , W^W^W , ... , W^W^W^
>    Thus he invents his famous series of the transfinite ordinal numbers.

I am afraid, but Cantor's invention was not quite so nonsensical as you
make it appear. It did NOT come from and idle mind: he DID have something
to count this way. He started from a set M=M^0 of real numbers on the
line. He defined

   M^n+1  to be set of accumulation points of M^n 

and then M^w as the intersection of all M^n . There are examples of sets
M such that M^w is not empty; hence the process can be repeated ... - and
this was what Cantor did. All of these sets are very concrete, and it all
was done to apply it to the exception sets of Fourier series. 

Also, in Schu"tte's Beweistheorie (the 1960 first edition) you find
examples in which the set of natural numbers is well ordered by very
high ordinal numbers - in one of them, for instance, 3 appears as omega
and 5 appears as epsilon-zero ...



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