Alexander Zenkin alexzen at
Tue Mar 30 19:12:31 EST 1999

  On  Tue, 23 Mar 1999, Walter Felscher (W.F.) wrote:

>  On Mon, 22 Mar 1999, Alexander Zenkin (A.Z.) wrote among many other

>>                                         But he <Cantor> disregards
>> these contradictions and absurdities, and boldly goes further:
>> the common operation "+1", defined for finite integers, to the
>> transfinite "NUMBER" W, he obtains, by means of, supposedly, his
>> 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^W , ┘ , W^W^W , ... , W^W^W^┘ (*)
>>    Thus he invents his famous series of the transfinite ordinal

Further, W.F. writes:

  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.

A.Z.- response:

   Sorry, but my words above are almost a literal citation from
G.Cantor. So, I can not change its sence.   As to M^w and the process
continuation, in the common classical mathematical analysis, there is
the "similar" theorem which states that any infinite system of embedded
linear segments, say, starting from a set M= M^0 = D_0 = [0,1]:

D_0,  D_1,  D_2,  . . . ,  D_n,  . . . ,   (1)

lengths of which are tending to zero, has a single common geometrical
point (real number), say,
d, or, that is the same, D_w which is, by Cantor, the "LAST" "segment"
in (1), or is "the
intersection of all D_n". If, as you say, "the process can be repeated",
then Cantor already comes
into the area of even not the "infinitesimals", but of the
"TRANSFINITESIMAL" things of the modern NON-standard analysis. As is
well known, G.Cantor was also afraid just actual
transfinitesimals panically and called them by "infinitesimal bacilli of
cholera in mathematics"
(Letters, 13 December, 1893). - Why? - Because he understood well that
he has no mathematical
competitors only in areas of TRANSFINITE LARGE INTEGERS, but, in the
area of
infinitesimals, any competition with Cauchy-Weierstrass' limits Theory
of Classical
Mathematics (and Mathematical Physics) was mortally dangerous to the
idea itself of his jump
into a TRANS-finite area, - "and this was  what Cantor could not do"

W.F. writes:

  All of these sets are very concrete, and it all was done to apply it
to the exception sets of Fourier series.

A.Z.- response:

   I think that all what has here a mathematical sense (as to the
Fourier series Theory) was obtained in a framework of Classical
Mathematics. Everything else refers to the area of wonders of
"Transfinite Paradise" which, according to D.Hilbert, was built "for us"
by G.Cantor.

W.F. writes:

     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 ...

A.Z.- response:

     You, Schu"tte, and many other recognized meta-mathematical experts
are absolutely right: there are thousand ways to "disembowel" the set of
natural number into any countable "very high ordinal". But so far as I
know, it is possible only with a distortion of the natural order of the
series of natural numbers. For example:

1,2,3,4,5, . . .  == > 1,2,4,5, . . ., 3, where "3 appears as omega".

    But "G.Cantor's Paradise" has many other wonders. Two of such new
wonders, that are not
known even in the modern meta-mathematics, I have presented and proved
in one of my previous messages (Subject: RE: As to Aristotel's
"Infinitum Actu Non Datur" Thesis; Date: Wed, 24 Feb 1999 03:43:3):

  1) there is a "very concrete" 1-1-correspondence between the series of
natural numbers,
   1,2,3, . . . (2)
and the series of G.Cantor's transfinite ordinals up to epsilon-zero
under the order preservation
in the both series; and

   2) there is an other "very concrete" 1-1-correspondence between the
series of natural numbers,
    1,2,3, . . . (2)
and the series of G.Cantor's transfinite ordinals up to any
epsilon-zero, epsilon-1, epsilon-2, and
so on under the order preservation in the series (2); and under the
elements quantity preservation
in the any given series of transfinite ordinals.

   The first, ORDER-ISOMORPHIC 1-1-correspondence shows that the true
mathematical (and ontological) sense of the famous series of G.Cantor's
transfinite ordinals up to epsilon-zero is not more then the
mathematical (and ontological) sense of the series of common finite
natural numbers.
    The second 1-1-correspondence shows that just the famous smallest
transfinite ordinal omega_1 (of the aleph_1 cardinality) is the first
unattainable ordinal. Indeed,
by definition, the omega_1 is a smallest transfinite ordinal which is
preceded by all COUNTABLE ordinals, but there exist neither a
meta-mathematician, nor an algorithm, nor a constructive criterion that
would be able to indicate a letter of the Grecian ABC (either EPSILON-W,
or DZETHA-W, or ETHA-W, and so on) which completes the construction of
the well-ordered set of "all COUNTABLE ordinals".

    Sorry, but together with Frege and Cantor, somebodies can only
regret (by some different reasons) that the Lord created the only
Paradise (without any logical and mathematical paradoxes), and the
Natural Numbers into it. "Alles andere ist Menschenwerk" according to
Leopold Kronecker together with Aristotle, Leibniz, Gauss, Cauchy,
Poincare, and many other outstanding thinkers who created the modern
Science (for more information see my previous messages and my WEB-Site).

Thank you for your very non-trivial objection.


 Prof. Alexander A. Zenkin,
 Doctor of Physical and Mathematical Sciences,
 Leading Research Scientist of the Computer Center
 of the Russian Academy of Sciences.
 e-mail: alexzen at
 "Infinitum Actu Non Datur" - Aristotle.

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