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

From: Plamen Petrov (ppetrov_at_hotmail.com)
Date: 10/14/04 

Date: Thu, 14 Oct 2004 12:57:49 +0200

Recently I had a private correspondence with Alexander Zenkin; among many
things I brought to his attention *this* thread discussing his publications
on Cantor.

I am forwarding the following message on behalf of Dr. Zenkin.

(You will find most of his comments below marked with "AZ" == "Alexander
Zenkin")

Yours truly, 

---
Plamen Petrov
http://digitalphysics.org
----- Original Message -----
From: Alexander Zenkin <alexzen@...>
To: <ppetrov@...>
Sent: Wednesday, October 13, 2004 1:44 AM
Subject: About the paradoxes in science and Zenkin's paper
>       To:Robert Low (mtx014@linux.services.coventry.ac.uk)
>       Re: Zenkin's paper on Cantor
>       Date: 2004-09-26 08:20:29 PST
>
> Eray Ozkural exa <erayo@bilkent.edu.tr> wrote:
> >Here is a much more refined attack on Cantor's proof than what has
> >lately occured in some threads that I have avoided:
> > http://www.com2com.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html
> >Does Alexander Zenkin have a good point?
>
> Robert Low responded:
> Maybe the one on the top of his head. For example, he
> seems to be under the impression that if you list all
> the finite expansions of the form
>
> 0.b1b2b3...bn
>
> where each bi is either 0 or 1, then there are only
> n of them.
> --
> Rob.  http://www.mis.coventry.ac.uk/~mtx014/
>
>  Dear Rob,
>
>  any crank knows well that "if you list all
> the finite expansions of the form
>
> 0.b1b2b3...bn
>
> where each bi is either 0 or 1, then there are" 2^n of such expansions,
> and no crank will use famous Cantor's Diagonal Method (further - CDM) in
> order to prove that 2^n > n.
>
>  I suspect that either you did nor read my paper or you know quite bad the
> elementary mathematics.
>
>  In my paper at
>
> http://www.com2com.ru/alexzen/papers/Cantor/Fatal_Mistake_of_Cantor.html
>
> is written quite cleary:
>
> "It is obvious that if the list of reals is finite, then its diagonal, and
> the corresponding (Cantor's)"anti-diagonal" sequence of binary digits
(say,
> 0.a1a2a3:an) will also be finite, i.e., in such a case it does not define
> any real number."
>  But if we shall write the digits of the "anti-diagonal" instead of the
> corresponding digits of ANY real from X=[0,1], e.g., 0.000 :, we shall get
a
> new real (even rational with the infinite 0-tail),
>
> 0.a1a2a3:an000:
>
> which, by its CDM-construction, is different from every real in the given
> finite list, i.e., which does not belong to this finite list.
>
>  Thus, the application of the famous CDM to infinite as well as to finite
> lists gives the same result: in the both of cases, the CDM allows to
> construct a NEW real, which is a real from X=[0,1] and is different from
> every real of the list to wich the CDM is applied. Consequently, from the
> point of view of the theory of algorithms, the method does not distinguish
> and does not take into account quantitative characteristics (properties)
of
> those sets and enumerations which it is applied to. 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 a number of their elements, i.e. by their
> "power-cardinality", does not differentiate (distinguish) finite sets from
> infinite sets just by their power!
>
>  Please, don't ascribe me your own doubtful fabrications and don't
disorient
> the readers with them.
>
>
>  Ryan Reich ryanr@uchicago.edu wrote:
>
> <Zenkin> discusses Cantor's proof, [:] using terms like "famous diagonal
> method".
>
> AZ:
> If you belive (in contrast to Hilbert, Kleene, Cohen, Boubaki and so on)
> that Cantor's Diagonal Method is not "famous", I shall not dispute with
you,
> since the Diagonal Method (further - DM) was well known to and was used by
> Pythagoras (in his DM-proof of the NON-finity of natural numbers) and
Euclid
> (in his DM-proof of the NON-finity of prime numbers).
>
> RR wrote:
> <Zenkin makes> "very general claims like "The one and only basis for the
> differentiation of such infinities is [Cantor's theorem]" without ever
> providing evidence.
>
> AZ:
> If you know (in contrast to Hilbert, Kleene, Cohen, Boubaki and so on)
other
> basis to differentiate infinite sets by their "cardinality" (unlike
Cantor's
> theorem), please let us know and the meta-mathematical community will be
> very grateful to you for ever.
>
> RR wrote:
> <Zenkin> "never says anything deep about the <Cantor> theorem or its
proof,
> but merely latches onto a single irrelevant feature which he perceives to
be
> of critical importance (the finite/infinite disparity) because he totally
> misunderstands set theory.
>
> AZ:
>  Yes, I never say some more deep then "Cantor's Theorem on the
> uncountability of continuum is unprovable, but its traditional "CDM-proof"
> is invalid". But I say nowhere that " the finite/infinite disparity " has
a
> "critical importance" in my disproof of Cantor, I say that
actual/potential
> infinitie disparity has a "critical importance" in order to understand
> Cantor's set theory. Unfortunately, even modern axiomatic set theory has
not
> a strict definition of these basic notions though "some axioms of
axiomatic
> set theory justificated by the acception of the actual infinity
conception"
> (S.Feferman) and "Cantor's diagonal argumen is based on the conception of
> actual infinity" (W.Hodges). See the correspoding discussion on the
FOM-site
> (FOM = Foundation Of Mathematics):
> A.A.Zenkin, AS TO STRICT DEFINITIONS OF POTENTIAL AND ACTUAL INFINITIES:
> http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html
> http://www.cs.nyu.edu/pipermail/fom/2003-January/006137.html
> Re: [FOM] As to strict definitions of potential and actual infinities.
> Alexander Zenkin
> http://www.cs.nyu.edu/pipermail/fom/2003-January/006173.html
>
> RR wrote:
>
> <Zenkin>  goes on to trot out the same argument that all the cranks do,
> namely that adding in the diagonal number must make the list complete
after
> all.  His justification demonstrates a lack of understanding of the nature
> of infinity and also of logic itself.  In fact, he says that the list is
> complete "at this moment" and then goes ahead and uses the diagonal
argument
> again, which indicates that he's heard the usual objections to this method
> but, having failed to understand them, he just sits on the fence and both
> accepts and denies them at once, hoping no doubt that when someone tries
to
> tear him down he can answer them no matter what their problem is.
>
>  AZ:
>  My proof is a strict isomorfic deductive model (in Tarski sense) of one
> well known invention. The invention is called the paradox of "Grand
Hotel".
> Whether the inventor of the paradox is crank as well? BZW, the "crank"
> D.Hilbert invented this paradox especially in order that childs and
freshmen
> could understand better the true nature of infinity. I recommend you
> insistently to be aware some better with this paradox before to terrorize
> readers of the site with your fatal misunderstanding "of the nature of
> infinity and also of logic itself".
>
> RR wrote:
> <Zenkin> "concludes his paper with the traditional appeal to authority,
> exhorting the reader to his point of view on the grounds that such
> luminaries as Aristotle, Leibniz, Cauchy, etc. all had personal dislikes
for
> infinite objects, as if this made much difference to the proof itself.
> Then, bizarrely, he closes with a reference to Freud."
>
>  AZ:
>  It is an everyday blatant lie "that all the cantorians and boubakists
do".
> At that, you know even history too bad. I could never and nowhere state
such
> a historical stupidity "that such luminaries as Aristotle, Leibniz,
Cauchy,
> etc. all had personal dislikes for infinite objects". I always stated and
> state that such luminaries as Aristotle, Euclid, Leibniz, Gauss, Cauchy,
> Hermite, Poincare, Bair, Borel, Lesbeg, Weyl, Luzin, and today Feferman,
> Peregrin, Vopenka, etc., etc., etc., liked "for infinite objects" very
much,
> but all they rejected categorically only the actual infinity and are, in
> reality, true creators of the true scientific theory of mathematical
> (potential) Infinity (don't mix up with famous Cantor's "theory" of
> NON-mathematical 'Transfinitum', based on the actual infinity conception).
>  BZW, I mention all these luminaries in the very end of my paper, it means
> that I don't use their authority in my disproof of Cantor's CDM-proof. And
I
> really wish to exhort the readers of my paper to the point of view that
the
> scientific intuition of these luminaries did not deceive them and their
> followers.
>
>  In a word, your complet distorstion, full misunderstanding of my paper
and
> a "zomby"-effect of you fine message to readers (hope, there are some
> "evidently nonsilly" students on the list) are well illustration to the
> following words of one of outstanding mathematicians and educational
> specialists of our time:
>
> "It is awful to think what kind of pressure the Bourbakists put on
> (evidently nonsilly) students to reduce them to formal machines! This kind
> of formalized education is completely useless for any practical problem
and
> even dangerous, leading to Chernobyl-type events. Unfortunately, this
plague
> of formal deduction is propagating in many countries, and the future of
the
> mathematics infected by it is rather bleak." - An Interview with Vladimir
> Igorevich Arnol'd by S. H. Lui. - Notices of the AMS, v.44, No. 4, 432-438
> (1997).
>
>  Read a special mathematical journals from time to time. It's useful to
know
> about what true (by Feferman) really working mathematicians think about
the
> faulty influence of modern bourbakism upon future generations, since, as
> Brouwer stated, "all Cantor' "Study on Transfinitum" is a pathological
casus
> in history of mathematics from which future generations will be
horrified".
>
>  AZ
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