(a fragment of my Letter to the Editor-in-Chief of the Bulletin of Symbolic Logic)
The Russell and Frege talented pupil and colleague, L.Wittgenstein was, contrary to W.Hodges' opinion, absolutely right when he stated that "Cantor's argument has no deductive content at all" (see [1], p. 6, and [12]). In reality, from the classical logic and classical mathematics point of view (see [7], [8], [10], [13]), the 10 strings of Cantor's "proof" of Theorem II prove the following (here B = "the enumeration (*) contains all real numbers of X").
1) The contradiction B à not-B of Cantor's proof has no relation to classical Aristotle's Logic at all.
2) The Cantor's proof itself is not a reductio ad absurdum proof, but it is a quasi-logical, i.e., pathological, version of the well-known counter-example method where, however, (in contrast to classical mathematics) a counter-example itself (the Cantor anti-diagonal number which is different from every number of the enumeration (*)) is deduced (!) logically and algorithmically from the non-authentic statement (assumption) B which then that counter-example itself "must" disprove.
3) The famous Cantor diagonal method which is a corner-stone of all modern meta-mathematics (as every philosopher knows well, all meta-mathematical proofs of famous Geodel's, Tarski's, Church's, etc. theorems are based just upon the Cantor diagonal method [9]) and the only ground, allowing meta-mathematicians and symbolic logicians to talk about that infinities are different in their cardinalities, does not distinguish infinite sets from finite sets just by their cardinalities.
4) In order to deduce the paradoxical implication [B à not-B], the Cantor diagonal method uses only the actuality property of the enumeration (*) and does not use the infinity property of that enumeration (*).
5) The formal "implication" [B à not-B] is a half of the famous "Liar", "Barber", etc. (here Y="Yes"):
[Y à not-Y]&[ not-Y à Y];
Of course, these paradoxes are not quarks of theoretical physics, but their "halfs" do not exist independently in the nature too.
6) Therefore explicitly using the infinity property of the same enumeration (*), we can construct the inverse paradoixical "implication" [not-B à B] and obtain a new set theoretical paradox:
[B à not-B] & [not-B à B] (1)
BTW this new paradox (in contrast to all known ones) arises not "in suburban areas of mathematics", according to Kleene [3,6] and Fraenkel and Bar-Hillel [9], but in the very core of the corner-stone of modern meta-mathematics.
7) In reality, the "process" (1) has the following potentially infinite form:
B à not-B à B à not-B à B à not-B à B à ... (2)
That correspondents well with the true potentially infinite form of the "Liar", "Barber", etc:
Y à not-Y à Y à not-Y à Y à not-Y à Y à ... .
The last was proved in [10] by means of the classical logic model theory.
8) It is easy to prove that the Cantor proof based upon the famous Cantor diagonal method is a deductive model (in A.Tarski's sense) of the standard Pythagorean proof that there are infinitely many natural numbers and "the standard Euclidean proof that there are infinitely many prime numbers" (see [1], p. 13). So the famous "Cantor diagonal method" was well known in classical mathematics from "the ancient Greeks time", but this ancient Diagonal Method was not leading ever to any paradoxes and great crises in foundations of mathematics because it was never applyed, before G.Cantor, to actually infinite sets.
9) The only reason of the paradoxality (2) is the actual infinity of the enumeration (*). It proves that "actual" and "infinite" are algorithmically contradictory notions in the framework of Cantor's proofs and, consequently, the notions "actual" and "finite" are here algorithmically identical ones. Thus, it proves first the Aristotle, Leibniz, Kant, Gauss, Cauchy, Kronecker, Poincare, Brouwer, Wittgenstein, Weil, Luzin, etc. intuitive opinion that the actual infinity notion itself is a self-contradictory notion.
10) Since every step of the infinite process (2) generates a new Cantor's anti-diagonal real number which is different from all already existing ones, and since there are not any logical and mathematical reasons to stop this process, it is a rigorous logical and algorithmical proof of not only the potnetial infinity of the set X, but also of the famous Aristotle Thesis.
As is well known, according to the ancient mathematical tradition, any rigorously proved mathematical statement is called a Theorem. So, taking into account all said above, we can write:
ARISTOTLE'S THEOREM (III B.C.). INFINITUM ACTU NON DATUR, i.e., the notion "actual infinity" is a self-contradictory one, and all infinite sets in true mathematics are potentially infinite.
In the very beginning of the XX Century, one quite well known expert exactly
in the field of the Mathematical Infinity, great Poincare warned, having in
mind all Cantor's "study on transfinitum", that "the actual infinity does not
exist; cantorians forgot about that and lapsed into the contradictions", and
that "one must not erect a scientific building on a sand" [11]. Poincare was
some inaccurate: the famous Cantor's "transfinite paradise" and all modern meta-mathematics
and non-naive axiomatic set theory were built not on a sand only, but
also on a blind, underhand, and tacit belief in the actual infinity,
i.e., in Cantor's Axiom, and in Cantor's transfinite "paradise" based exclusively
and completely upon that Axiom.
The question arises about how so many people with a quite high IQ, of course,
including symbolic logicians not only "alive today", were able to have so deep
mystic belief in the mythic Cantor's "paradise" during all the so advanced XX
Century? – I think that the question will be a non-trivial and very important
social problem for philosophy of science and psychology of cognition in the
XXI Century.
Defining the already today's harmful scientific, educational, and
social consequences of the "left-hemispherical bourbakization" of mathematics,
the Vice-President of the International Mathematical Union, V.I.Arnol'd, writes
(see [14] and my comments in [15]):
"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".
So, in order to the, by Gauss, "Queen of all sciences" could, at last,
restore the order in Her possessions, disturbed by the Third Great Crisis in
foundations of mathematics, I offer that the International Mathematical Union,
in the person of its today's respected commanders (D.Mumford, V.Arnold, W.Thurston,
Yu.Manin, etc.), to oblige modern meta-mathematics, symbolic logic, and
non-naive axiomatic set theory commanders to include, at last, according to
the generally accepted standards of classical mathematics, the Cantor Axiom,
as the only necessary condition of all epochal achivements of all mentioned
sciences, in all axiomatic set theoretical systems and textbooks explicitly
in order to everybody can see the true "foundation" of famous Cantor's "paradise".
All the rest will be made fine by really well and wide thinking "mathematicians
alive today", and also by classical logicians, philosophers, and, of course,
psychologists.
BTW, as is known Cantor's "paradise", according to Brouwer and his intuitionism
school, is “a curious pathological casus in history of mathematics by which
further generations will be ... horrified” [9]. It is really so, and I suggest
a unique WEB-Project, say, "Aristotle-XXI" aimed to preserve these future generations
from that "horrors" by common efforts of those mathematicians and philosophers
who are, so far, not too infected by the dangerous near-scientific illness
of the XX Century - the left-hemispherical para-"mathematical"
bourbakization of the mind.
All said (and much more not said) above is a first step to realize the
Project.
Regards W.Hodges and other "teachers" in the "basic elementary logic", I
am forced to remind them of Bertrand Russell's letter-1902 to G.Frege. The Russell
"Barber" was not only an ingenious, but regular paradox, it was not only the
first strike of the bell which heralded the beginning of the Third Great Crisis
in foundations of meta-mathematics. It was an Omen from on High: "The
road to Cantor's "paradise" is paved not only "with good intentions", but also
with contradictions and paradoxes. Don't create a new "Babel"!" – But a few
desired to hear and understand the true sense of that Omen.
At any rate, it was not the Lord who suggested to G.Cantor and his followers
"to build a transfinite stairs to the heavens". As is well known, the First
Babel was destroied by the Lord. I believe, the transfinite Babel created by
a human-being's unlimited and unwise proud and ambitions only will suffer the
same fate... Alas.
I would like to complete these comments by the (proved above) Aristotle's Providence:
long before Georg Cantor, today, and for all time,
INFINITUM ACTU NON DATUR.
REFERENCES
[1] W.Hodges, An editor recalls some hopeless
papers, The Bulletin of Symbolic Logic, vol. 4, Number 1,
March 1998
[2] N.Bourbaki, Set Theory. – Moscow : "MIR", 1965
[3] S.Kleene, Introduction to metamathematics. - Moscow :
MIR, 1957.
[4] Marek Capiński and Ekkehard Kopp,
Measure, Integral and Probability.- London: Springer-Verlag
London Limited 1999, pp. 4-5
[5] Handbook of Mathematical Logic,
Jon Barwise (Ed.). – North-Holland Publishing Company, 1977. Part II. Set Theory.
[6] S.Kleene, Mathematical
Logic. – Moscow: MIR, 1973.
[7] A.A.Zenkin, George Cantor's
Mistake. - Voprosy Filosofii (Philosophy Problems),
2000, No. 2, 165-168.
[8] A.A.Zenkin, Fatal Mistake of
G.Cantor's Theory. – Some advanced version of [7] at:
http://www.com2com.ru/alexzen/vf1/vf-eng.html
(in English).
http://www.com2com.ru/alexzen/vf1/vf-rus.html
(in Russian).
[9] A.A.Fraenkel, Y. Bar-Hillel, Foundation of Set Theory.
– North- Holland Publishing Company, Amsterdam, 1958.
[10] A.A.Zenkin, New Approach to the Paradoxes Problem Analysis.
- Voprosy Filosofii (Philosophy Problems), 2000,
No. 10, 79-90.
[11] A.Poincare, On Science. - - Moscow : SCIENCE, 1983.
[12] L.Wittgenstein, Remarks on the foundations of mathematics.
- Blackwell, Oxford, 1956.
[13] A.A.Zenkin, Cognitive (Semantic) Visualization Of The Continuum
Problem And Mirror Symmetric Proofs In The Transfinite Numbers Theory. -
The e-journal "Visual Mathematics" at the WEB-Sites:
http://www.mi.sanu.ac.yu/vismath/zen/index.html
(in English).
http://members.tripod.com/vismath1/zen/index.html
(in English).
[14] An Interview with Vladimir Igorevich Arnol'd by S. H. Lui. -
Notices of the AMS, v.44, No. 4, 432-438 (1997).
[15] A.A.Zenkin, "Scientific Counter-Revolution in Mathematics".
- "NG-SCIENCE", Supplement to the "Independent Newspaper" (Nezavisimaya
Gazeta) on 19 July, 2000, pp. 13. At the WEB-site:
http://science.ng.ru/magnum/2000-07-19/5_mathem.html
(in Russian)
http://www.com2com.ru/alexzen/papers/ng-02/contr_rev.htm
(in English)
[16] P.S.Alexandrov, Introduction to Common Theory of Sets and Functions.
- Moscow-Leningrad : Gostehizdat, 1948.
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