Re: Zenkin's paper on Cantor (reply of Dr. Zenkin)

From: tinyurl.com/uh3t (rem642b_at_Yahoo.Com)
Date: 12/21/04 

Date: Mon, 20 Dec 2004 16:12:49 -0800

> From: "Plamen Petrov" <ppetrov@hotmail.com>
> http://www.com2com.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html
Sometimes the English is not quite standard, but:
> e-mail: alexzen@com2com.ru .
> Voprosy Filosofii (Philosophy Problems), 2000, No. 2, 163-168.
I would guess English is not the author's native language. Accordingly
where I criticize the English usage below, please don't take it as
chastisement, merely as feedback to help you clean up your paper to be
better English and thus more easily understandable.

   ... the problem of the foundations of that sets theory ...

It's usually called "set theory" (without the "s").

   Modern (axiomatic) theory of sets is the only mathematical
   discipline, which "knows how" to differentiate infinite sets one from
   another according to their powers, i.e., basing on the number of
   elements making up that set.

"powers" is a poor choice of word. Why not use the word "cardinality"?
"power" in set theory usually refers to the power set or something
related to it, so the word is misleading here.

"basing" should read "based".

   The one and only basis for such
   differentiation of infinities is the famous George Cantor's theorem
   about the uncountablity of the set of all real numbers.

No no no. The one and only basis for differentiation of infinities on
the basis of size is 1-to-1 mappings between two sets or between a set
and a subset of another. If there is such a mapping between two sets,
they are of the same cardinality. If there is such a mapping between
one set and a subset of the other, but not vice versa (second set and
subset of first set), then the former set is "smaller" than the latter.
Cantor's theorem about uncountablity of real numbers is merely one tiny
application of that methodology, not the basis of the methodology.

   binary digit G.Cantor, using the diagonal (*), builds a new infinite
   binary sequence: ...
   according to his famous diagonal rule:
       for any i: [if x[ii ]= 0, then y[1i ]= 1] and [if x[ii ]= 1, then
   y[1i ]= 0]. (***)

Ignoring the problem with the first subscript being 1 again, using
binary digits has a well-known problem that one two different
representations of the same number may appear, one in the original list
and one as the newly-constructed anti-diagonal number. To avoid that
flaw, you need to use a base larger than one, either explicitly, or by
grouping the binary digits somehow.

   the inapplicability of Cantor's diagonal
   method to finite enumerations is so obvious that nobody ever examined
   that issue.

It's not a problem! Just as an infinite enumeration (mapping from
natural numbers to reals) produces an anti-diagonal not among the
enumeration hence the enumeration couldn't have been all the reals, a
finite enumeration produces an anti-diagonal not among the enumeration
etc. But we already knew the reals weren't a finite set, so the second
(finite) case can be eliminated from the very start.

   Thus, Cantor's diagonal method proved to be applicable, without
   any changes, to both infinite and finite enumerations.

Correct.

   We come to the following, very
   significant for the mathematics philosophy conclusion: The only
   method, which hitherto allowed meta-mathematicians to differentiate
   sets according to the number of their elements, i.e. by their
   "power-cardinality", does not differentiate (distinguish) finite sets
   from infinite sets just by their power!

Nope. The way to distinguish the size of two sets, as I said before,
two sets can be put into 1-1 correspondence with each other, and if not
then by checking which can be put into 1-1 correspondence with subset
of other. For example a set can be compared with the set of natural
numbers to find whether it's smaller (hence finite), same size (hence
countably infinite), or larger (hence uncountably infinite). Cantor's
diagonal argument is merely a tool to prove that it's impossible for
certain types of 1-1 correspondences to exist.

   It's obvious that it is only the actuality of those enumerations
   to which the method is applied, ...

At this point the text starts making no sense whatsoever.

   REMARK 2. Using the second popular version of Cantor's proof by
   assuming the existence of a 1-1-correspondence between infinite set,
   say U, and the set, P(U), of all its subsets (see for instance
   Hausdorff F., Theory of Sets. M.-L.: ONTI, 1937, p. 33-34) equally
   does not change anything, ...

That's a different theorem, but indeed the same kind of logic applies:
Assumption that U and P(U) are in 1-1 correspondence produces a
contradiction, thereby proving there is no such correspondence.
And all your nonsense regarding enumeration of reals (starting where I
said "at this point .. no sense whatsoever"), converted to apply to
enumeration of power set, likewise remains nonsense.

As to your claim that infinite sets are impossible: Well then the
natural numbers are impossible to you, and therefore you can't even
contemplate that kind of simple arithmetic. So questions whether an
infinite set, which you don't believe can exist in the first place, are
countable or not, are moot to you, and you have no business posting
anything about them. By the way, if you don't believe there are an
infinite number of natural numbers, then please tell us which natural
number you (Alexander Zenkin, or Voprosy Filosofii) believe is the
largest of them all.

Relevant Pages

  • Re: Zenkins paper on Cantor (reply of Dr. Zenkin)
    ... I would guess English is not the author's native language. ... Just as an infinite enumeration (mapping from ... to both infinite and finite enumerations. ...
    (sci.math)
  • Re: An uncountable countable set
    ... How could an enumeration by natural numbers supply ... and when you stop you have done a finite number of transpositions you ... the second sequence of transpositions. ... the meaning of an infinite sequence of transpositions. ...
    (sci.math)
  • Re: An uncountable countable set
    ... If an enumeration rule exhausts the naturals, ... That does not make their sizes infinite. ... of transpositions you give has order type w * w. ...
    (sci.math)
  • Re: Cantors wrong! (doubt it)
    ... > finite length and exactly 1 real of infinite length, not all reals. ... > presented in contradiction to cantor is invalid when it compares an ... > enumeration of integers because it only shows that an integer HAS not ...
    (comp.theory)
  • Re: just 5 quick answers then I can summarise and GO
    ... > Herc> There are infinite stars out tonight. ... If it's intended to be English ... There are infinite stars in the sky. ...
    (sci.logic)
Loading