ubject:
As to a "naivete" of G.Cantor's set theory
Date:
Sun, 24 Jan 1999 15:49:15 +0300
From:
Dear Colleagues,
Thanks very much to all who have taken a part in the discussion on a
sense of the "naive" theory term.
I would like to make some remarks too.
1. AS TO INFORMAL = "NAIVE", FORMAL = NON-"NAIVE" CRITERIONS.
It is quite strange criterion. The Euclidean axioms system for planar
geometry (III B.C) is the first example of a perfect formal system. So,
it is non-"naive". The modern mathematical physics "systems", which
solve brilliantly any problems of modern advanced natural science and
industry (XX C.), is not formal (in the meta-mathematical sense), so it
is "naive" theory!? In such the case, all modern, say, mathematical
analysis, mathematical physics, analytical number theory, and so on, are
"naive" theories!? But I am sure that none of professionals in that
areas of modern mathematics will be consenting to such the "naive"
definition and valuation of his/her activity area.
Of course, the problem is much deeper. In my opinion, there are, at
least, two principal differences between meta-matematical formal systems
and physical theories (in the form of their formal mathematical models).
1) In a framework of a meta-mathematical formal system, there is the
only criterion of the truth: the correct deductive inference of a
statement from an axioms system. In a framework of a mathematical model
of physical theories (a system of differetial equations (axioms) and
mathematical (deductive) rules of their formal transformations) it is
not sufficient: your deduction can be correct, but if a mathematical
deductive inference is not coincided with the picture of corresponding
real physical phenomen then your deduction, and your formal model (i.e.,
mathematical theory) are incorrect. That is in real mathematical formal
systems there are the two truth criterions: the formal-deductive one
(that is necessary, but not sufficient), and an absolute one (the
compliance with experiment) which has no place in modern
meta-mathematics.
2) If you add, for example, the Continuum Hypothesis or its negation
to, say, ZF-system, you can in both cases state a question on the
truth/falsity of deducible statements. Now, try to add, say, the
Shreodinger's equation (it is the most formal and most rich axiom of the
modern quantum mechanics) to, say, the formal mathematical model of the
classical Newton's mechanics. Obviously, the question itself what
inferences in a framework of such an "advanced" formal system will be
true/false sounds not very interesting from the formally mathematical
and experimentally physical points of view.
2. PEDAGOGICAL PROBLEM. The term "naive" as to a scientific theory has
an obvious disparaging sense (see, for example, Halmos' citation "
general set theory is pretty trivial stuff really, but if you want to be
a mathematician, you need some, and here it is; read it, absorb it, and
forget it.", given in Charles Silver's message of Mon, 25 Jan 1999
14:24:26), therefore if a recognized professor calls "naive" a theory,
then none of his students will wish to wast time to learn such a "naive"
theory. As a result, such students will never know those ideas,
thoughts, noble creative impulses and so on, which transformed, say, a
g.cantor into the famous G.Cantor. After 2-3 students generations,
already new generation of recognized professors will not be knowing
about that, not saying already about their today's and future students.
3. THE MAIN G.CANTOR'S IDEA. About what did G.Cantor constantly think? -
About one ETERNAL problem only: whether it is possible and how to turn
the fuzzy, vague, "philosophical" notion of the actual infinity to a
logically legitimate mathematical object and how to prove its
consistency (the last is not made up today).
But what have we today? The words "actual infinity" (not saying
already on "potential infinity") are simply banished from the modern
meta-mathematics dictionary. For example, "From the axiomatic point of
view, - A.A.Fraenkel and Y.Bar-Hillel write in the unique, brilliant
"Foundations of set theory", 1958, Chapter II, Paragraph 5, - there is
only one possibility of obtaning infinite sets - to postulate their
existence". Best of all the N.Bourbaki solves that non-"naive" problem
(see N.Bourbaki, "Theorie des ensembles", 1958, Chapter III, Paragraph
6.1):
"A5. There exists an infinite set."
What kind of the Infinity does it mean, actual or potential one? -
There is none word about that for the space of about 500 pages. It is
obvios that the question is not about the potential infinity. But maybe,
modern meta-mathematics does not use such the doubtful objects as actual
infinite sets too, which many scientists consider as a main reason of
all paradoxes and contradictions in the foundation of mathematics? - Let
any infinity set be a potentially infinite set, and all set-theoretical
paradoxes will vanish (of course, together with the modern high skill to
distinguish infinite sets by their common, large and unaccessible
cardinalities).
4. ALL MODERN META-MATHEMATICS IS STRONGLY BASED UPON THE NOTION OF THE
ACTUAL INFINITY.
Since any axiomatic set theory, without G.Cantor's Theorem on the
uncountability, is not a very interesting theory, and since that
G.Cantor's Theorem itsef becomes invalid and loses its sense without the
actual infinity, it can be stated that all modern meta-mathematics is
strongly based upon just such the "naive" notion as the ACTUAL infinity.
5. AS TO G.CANTOR'S THEOREM ITSELF, not all is clear here even today. I
have in my mind the following counter-example.
Let X be the set of all real numbers of the segment [0,1]. Introduce
the following notation:
1) For simplicity, we shall use the binary number system for the
representation of real numbers x of X.
2) We define {L:}, where "L:" is a symbol label for a statement
directly following the right brace "}".
Consider a traditional proof of G.Cantor's Theorem on the
uncountability of the real numbers set X.
G.CANTOR'S THEOREM: {Thesis A:} The set X is uncountable.
G.CANTOR'S PROOF. Assume that {Assumption not-A:} the set X is
countable.
Then there is an enumeration of all x (- X. Let
x1 , x2, x3 , . . . xn , . .. (1)
be such a countable enumeration of all x (-X, i.e.,
{ B:} for any x, if x (- X then x (- (1).
Applying his famous Diagonal Method to the countable enumeration (1),
G.Cantor constructs a new "diagonal" real number (DRN), say, y1, such
that, by definition, y1 is an element of X, but, by the very
construction, y1 differs from every element of (1), i.e., y1 does not
belong to (1). Consequently,
{not-B:} the DRN y1 (- X, but the DRN y1 does not belong to (1).
Thus the desired contradiction (between not-B and B) is obtained.
Here G.Cantor makes his famous conclusion: the assumpution not-A is
false, thus Thesis A is true, Q.E.D.
Unfortunately, G.Cantor's Diagonal Method (CDM) does not take into
account that the enumeration (1) is not only an actual set containing
all x of X, but also (1) is an infinite set, and CDM does not use the
last property of the enumeration (1) (for more info see [1-5] below). As
is known, according to Cantor's own definition of the infinite set
notion, an infinite set cardinality does not change if we add to it a
single element. Therefore, we can add the DRN, y1, - which is a single
DRN for the enumeration (1) by virtue of its binary representation, - to
the infinite enumeration (1), for example, by such the way:
y1 , x1, x2 , x3 , . . . xn , . (1.1)
Obviously, the countable infinite enumeration (1.1) contains now all
real numbers of X, i.e.
{ B:} for any z, if z (- X then z (- (1.1).
So, we have eliminated the contradiction between not-B and B, without
any contradiction with the assumption, not-A, about the countable
infinity of the initial enumeration (1). Now, again applying G.Cantor's
Diagonal Method to the countable infinite enumeration (1.1), we can
construct a new DRN, say, y2 such that,
{not-B:} the DRN y2 (- X, but the DRN y2 does not belong to (1.1).
Further, we construct a new countable enumeration, say such:
y2 , y1 , x1, x2 , x3 , . . . xn , . (1.2)
and have
{ B:} for any z, if z (- X then z (- (1.2).
But constructiong a new DRN, say, y3, we have
{not-B:} the DRN y3 (- X, but the DRN y3 does not belong to (1.2).
Then, for the new countable enumeration, say, such:
y3 , y2, y1 , x1, x2 , x3 , . . . xn , . (1.3)
we will have:
{ B:} for any z, if z (- X then z (- (1.3).
And so on.
It is obvious that G.Cantor's proof itself is transformed into the
following form:
not-A == > B== > not-B == > B == > not-B == > B == > . . . (2)
Obviously, there exist neither logical, nor mathematical reasons in
Nature which could allow us to stop the infinite proccess (2). Also
obviously, that we will never obtain any contradictions with G.Cantor's
assumption, not-A, about the countable infinity of the initial
enumeration (1).
Taking into account the fact that a new DRN is constructed at every
step of the proccess (2), we can conclude.
1. The G.Cantor's proof involves, in reality, the non-finite stage (2),
and thus that proof is not a mathematical proof in the well-known
D.Hilbert's sense.
2. G.Cantor's final conclusion about the uncountability of the set X
"vaults over" the potentially infinite stage (2); i.e. the traditional
proof by G.Cantor involves the fatal logical mistake "jump to a
<wishful> conclusion".
3. G.Cantor's "proof" actually proves just the potential character of
the infinity of the set X.
REMARK 1. The use of any other radix of a number system >2 changes
nothing, i.e., it leads to the same conclusions 1-3.
REMARK 2. The use of the second known variety of G.Cantor's proof (by
way of an assumption on the existence of an 1-1-correspondence between
an infinite set,U, and a set, P(U), of all its subsets (see, for
example, Hausdorff, F., Set Theory. - M.-L.: ONTI, 1937, pp. 33-34 (in
Russian)) changes nothing, i.e., it leads to the same conclusions 1-3.
A complete analysis of other mistakes of G.Cantor's proof of the
uncountability theorem is given in [1-5].
REFERENCES
1. A.A.Zenkin, New paradox of Cantor's Set Theory. - International
Conference "V.A.Smirnov's readings", Moscow, 18-20 March, 1997. Section
1. Symbolic Logic. Abstracts, pp 17-18. (in Russian).
2. A.A.Zenkin, The Time-Sharing Principle and Analysis of One Class of
Quasi-Finite Reliable Reasonings (with G.Cantor's Theorem on the
Uncountability as an Example) - Doklady Mathematics, vol 56, No. 2, pp.
763-765 (1997). Translated from Doklady Alademii Nauk, Vol 356, No. 6,
pp. 733 - 735.(1997).
3. A.A.Zenkin, Cognitive Visualization of some transfinite objects of
the Classical Cantor Set Theory. - In the collective monography
"Infinity in Mathematics: Philosophical and Historical Aspects", Edr.
Prof. A.G.Barabashev. - Moscow: "JANUS-K", 1997, pp. 77-91, 92-96,
184-189, 221-224. (in Russian).
4. A.A.Zenkin, On Logic of Some Quasi-Finite Reasonings of Set Theory
and Meta-Mathematics. New Paradox of Cantor's Set theory. - News of
Artificial Intelligence, 1997, no.1, 64-98. (in Russian).
5. A.A.Zenkin, Whether the Lord exists in G.Cantor's Transfinite
Paradise? - News of Artificial Intelligence, 1997, No. 1, pp. 156-160.
(in Russian).
Alexander Zenkin
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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 com2com.ru
WEB-Site http://www.com2com.ru/alexzen
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"Infinitum Actu Non Datur" - Aristotel.