Proceedings of the SYMMETRY FESTIVAL 2003
where Science Meets Art, 18-24 August, 2003, Budapest , Hungary.
http://www.com2com.ru/alexzen/papers/CANTOR-2003/ZENKIN%20BUDAPEST.doc
ONTOLOGY OF MIRROR SYMMETRY IN LOGIC AND SET THEORY
AS A WAY TO SOLVE THE FIRST HILBERT'S PROBLEM.
Alexander ZENKIN
Computing Center of the RAS, Vavilov st. 40, 117967 Moscow GSP-1, Russia.
mailto: alexzen@com2com.ru
Keywords: mirror symmetry, foundations of mathematics, incompleteness of Cantor's set
theory, solution of continuum problem, logic and philosophy of science, ontology.
In 1900, at the Second International Congress of Mathematicians in Paris, Hilbert
formulated his famous 23 open mathematical problems which were, by his opinion, most
important for a progress of Mathematics in the coming century. And the First problem of this
famous list was Cantor's Continuum Hypothesis (CH). Cantor first stated the CH in 1878
(Cantor, 1985). The sense of the CH consists in the following. Let X=[0,1] be a set of all real
numbers (alias of all proper fractions, alias of all points) of the segment [0,1], N be the set of
all finite natural numbers of the infinite series,
1,2,3, ..., n, ... ,
(*)
and the notation |Y| be a number of elements (alias power or cardinality) of the set Y for any
infinite set Y. So that |N| = 0 is, by definition, a minimal infinite cardinal number, and |X| =
C is a power of continuum. Using his famous theorem on the cardinality of powers-sets,
Cantor constructed the series of transfinite cardinal numbers (powers):
0 , 1 , 2 ,..., n , ...,
(**)
where 1 is a minimal uncountable cardinal, and formulated the CH in the following form.
CANTOR'S CONTINUUM HYPOTHESIS: C = 1, where С = 20 > 0 .
K.Goedel (1939) and P.Cohen (1962) proved the independence of the CH within the
framework of the axiomatic set theory of Zermelo-Fraenkel, but the proof of the CHindependence and the CH-solution are obviously quite different things. The situation is
described best of all by P.Cohen himself. Concerning a solvability of the CH by means of
modern meta-mathematical and set-theoretical methods, Cohen writes in the very beginning
of his known book (Cohen, 1969, p. 13): "... Continuum Hypothesis is a rather dramatic
example of what can be called (from our today's point of view) an absolutely undecidable
assertion ...". And completes the book by the words (Cohen, 1969, p. 282): "Thus, C is greater
than n, , , where  = , and so on. From this point of view, C is considered as an
incredibly large set, which is given to us by some new bold axiom and to approach to which
by means of any gradual process of construction is impossible." Taking into account such the
quite pessimistic opinion of the leading expert in this area as to a solvability of the CH, we
shall use the following more weak, but much more general formulation of the CH or, more
precisely, - of the Continuum Problem (CP) as a whole (Zenkin, 2002, 1999, 1997).
CONTINUUM PROBLEM FORMULATION. Is there a set of integers, say D, such that a
one-to-one correspondence between the sets, D and X, can be realized so that |D|=|X|=C?
Consider the well-known in mathematics binary tree (Fig.1.) that is a cognitive visual
representation (image) (Zenkin, 1991) of the set X alias the continuum: any infinite path,
V a1 a2 a3 ... an ..., where i1 [[ai =0] or [ai =1]],
(1)
of the tree represents an only real xX, and any real xX (2) defines an only path of the tree.
So, there is a 1-1-correspondence between X and a set, say P, of infinite paths of the tree.
1
Proceedings of the SYMMETRY FESTIVAL 2003
where Science Meets Art, 18-24 August, 2003, Budapest , Hungary.
http://www.com2com.ru/alexzen/papers/CANTOR-2003/ZENKIN%20BUDAPEST.doc
Since our Symmetry-Festival is devoted to
interdisciplinary and holistic approaches in
different areas of a human-being's activity, it's
appropriate here to demonstrate the following
unexpected connection between the pure
mathematical object presented in Fig.1 and the
most ancient holistic conceptions of
Humankind as to System of Universe.
According to modern mathematical graph
theory, the graph shown in Fig.1 is really called
a tree, the vertex V of the graph is called a root
of the tree, and the main topological property of
the graph consists in that any two infinite paths
of the tree have no common points except for
the root V.
In the far 70s of the last century, when the
(elder) author was investigating the CP and all
his writing-table was swamped with papers,
manuscripts, and pictures like Fig.1, a book about ancient Indian philosophy turned out at the
table, absolutely by accident. The book was opened at a page, by accident. The page was
saying: "Its roots are at the top, its branches are at the bottom - such is the ancient fig Tree. It
is Truth, indeed. It is Brahman. All the worlds are based upon it.... " [Cognition of Brahman.
Katha-Upanishada, II, 3].
Whether already the ancient Indians knew well the Tree and were seeing well that the
segment [0,1] is a mathematical model of continuity, which is one of the most important,
basic concepts of mathematics which, according to Gauss, is "a Queen of all sciences" with
the help of which modern humankind tries to comprehend a veritable sense of the Universe?!
Now denote the tree in Fig.1. by TR , rotate it counter-clockwise at 90, and place a mirror
AB at its root V in parallel to its levels. Then, we can see the visual result of such the
transformation in Fig. 2, that demonstrates a cognitive visual image of the mirror-symmetrical
(MS) 1-1-correspodence, say , between the original tree TR and its mirror image - the tree
TL . The math interpretation of the levels of the trees TR and TL is also shown in Fig. 2.
Consider a real number, x  [0,1], written in binary system, and its equivalent notation in
the form of the corresponding infinite sum:

x = 0.a1 a2 a3 ... an ...
 x =  a i 2  i ,  i1 [[ai =0] or [ai =1]].
(2)
i=1
In connection with the MS-mapping  there are interesting only two following cases.
CASE 1. Let x be a rational number with a 0-tail, i.e., in (2) ki>k [ai =0]. Denote the set
of such numbers as Q0. By , this x is associated with an integer x possessing the property:

k
x = ...a k ...a 2 a1a 0   a 2i   a 2i   , so that x is a finite natural number, (3)
i=0 i
i=0 i
and we have |Q0| = |N| = 0.
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Proceedings of the SYMMETRY FESTIVAL 2003
where Science Meets Art, 18-24 August, 2003, Budapest , Hungary.
http://www.com2com.ru/alexzen/papers/CANTOR-2003/ZENKIN%20BUDAPEST.doc


x =  a i  2 -i
x =  a i  2i
i 0
i 0
TL
TR
A
B
a)... ... n ...
b)...2 ... 2n ...
c)... a  ... a n ...
4 3
24 23
2
22
1
21
a4 a3 a2 a1
0


1
2-1
2 3 4
2-2 2-3 2-4
a1 a2
a3 a4
... n ...  ...
... 2-n ... 2- ...
... an ... a ...
Fig.2. Cognitive Visualization of the Continuum Problem:
a) level's numbers of the trees; b) powers of the base 2 in the binary system;
c) binary representation of the “in-both-side transfinite” hyper-real numbers.
 A.Z., 1975
CASE 2. Let x be an irrational (or rational with a non-zero tail) number, i.e., in (2) ik>i
[ak = 1]. Denote the set of such numbers as D. Then, by virtue of , this x is associated with
an integer x with the following property:

(4)
x = ...a n ...a 2 a1 a 0 =  a 2i = S,
i=0 i
By math analysis, the infinite sum S is a limit of the infinite sequence of its partial sums,
n
s1,s2,s3,. . .,sn,. . ., where sn =  a 2i < , so that sn is a finite natural number.
(5)
i=0 i
It is obvious that (5) is an infinite subsequence of (1). The main properties of this new
mathematical objects are given by the following statements (Zenkin, 1999, 1997).
MS-THEOREM 1. For all x D the ordinal type of the transfinite integer x is equal to
Cantor's minimal transfinite ordinal number , i.e., Ord( x ) = .
MS-COROLLARY 1. Any infinite subsequence of the natural series (*) is an individual
mathematical object which is a transfinite ordinal 'integer' of the Cantor's -type.
MS-COROLLARY 2. For any two real numbers x1, x2  X, if x1x2 then x  x for associated transfinite integers x , x  D.
1
1
2
2
The MS-mapping gives us a unique opportunity to produce absolutely rigorous
ontological, but conditional inferences concerning the existence of such transfinite objects.
'IF-THEN' MS-THEOREM 1. IF a geometrical point x[0,1] is an individual object THEN
the corresponding infinite path (1) of the tree TR arrives at its -level.
COROLLARY 1. All infinite (1)-paths of the tree TR are actual and arrive at the -level.
'IF-THEN' MS-THEOREM 2. IF an infinite (1)-path of TR arrives at the -level THEN the
-associated infinite (4)-path x of TL arrives at the -level of the tree TL as well.
3
Proceedings of the SYMMETRY FESTIVAL 2003
where Science Meets Art, 18-24 August, 2003, Budapest , Hungary.
http://www.com2com.ru/alexzen/papers/CANTOR-2003/ZENKIN%20BUDAPEST.doc
COROLLARY 1. All infinite (4)-paths of the tree TL are actual and arrive at the -level.
'IF-THEN' MS-THEOREM 3. IF a (4)-path x of the tree TL arrives at the -level THEN
Cantor's ordinal number of the transfinite integer x is equal to , i.e., Ord{ x }=.
COROLLARY 1 By , Card{all x TL } = Card{all xTR }, i.e. |D|=|X|=C.
'IF-THEN' MS-THEOREM 4. IF the geometrical point x of the segment [0,1] exists as an
individual object THEN there exists the Cantor least transfinite integer  and the infinity of
all sequences (1) - (5) are thus actual.
The theorem 4 is a strict (unlike Cantor) conditional MS-proof of the Cantor's  existence.
Thus the new transfinite mathematical objects (4) have the same ontological status as the
usual irrational numbers (2). It is quite appropriate here to remind the well-known oracular
G.Cantor's words: “Transfinite numbers themselves are, in a certain sense, new
irrationalities.[...] It can be certainly said: transfinite numbers stand and fall together with
finite irrational numbers.” (Cantor, 1985). All these conditional MS-statements are valid only
iff all infinite sets are, according to Cantor, actual. As to a strict algorithmical definition of
the concepts of potential and actual infinity see the [FOM]-message at
http://www.cs.nyu.edu/pipermail/fom/2002-December/006121.html. An artistic dynamical
cognitive movie "A colour-musical "black hole" of the continuum problem" will be
demonstrated during our report (Zenkin, 2001, 1991).
MAIN CONCLUSION. Let  be a set of all countable ordinals. According to Cantor, ||
= 1. On the other hand, the set x of all transfinite -ordinals is a proper subset of .
Consequently, | x | = C  || = 1, and Cantor's CH has thus the following solution: C  1.
However, the proven fact that there is not the only Cantor's ordinal , but the uncountable set
of different ordinals of -type means that all Cantor's theory of transfinite ordinals is
informally incomplete, is based on a doubtful semantics and must be essentially revised as a
whole. It should be emphasized especially once more that all the conditional ontological MSTheorems above are strictly valid iff the Cantor's very non-naïve axiom "all infinite sets are
actual" holds (Zenkin, 2002).
References
Cantor Georg, "Proceedings in Set Theory". - Moscow: NAUKA, 1985 (in Russian).
Cohen, Paul J. Set Theory and the Continuum Hypothesis. - Moscow : MIR, 1969 (in Russian).
Zenkin A.A., Cognitive computer graphics. - Moscow : "Nauka",1991. See Synopsis at:
http://www.com2com.ru/alexzen/.
Zenkin A.A., Scientific Intuition Of Genii Against Mytho-"Logic" Of Transfinite Cantor's Paradise. International
Symposium "Philosophical Insights into Logic and Mathematics ", 2002, NANCY, France.
http://www.philosophy.ru/library/math/sci_intuition.pdf
Zenkin A.A., Goedel's numbering of multi-modal texts. - The Bulletin of Symbolic Logic, Vol.8, No.1, 2002, p.
180.
Zenkin A.A., Cognitive (Semantic) Visualization Of The Continuum Problem. - "Visual Mathematics", Volume
1, No. 2, 1999. at the WEB-Sites: http://members.tripod.com/vismath1/zen/index.html
Zenkins Alexander and Anton, Presentation "The Unity of the Left-Hemispheric, Rational, Abstract Thinking and
the Right-Hemispheric, Intuitive, Visual One. Intellectual Aesthetics of Mathematical Abstractions". – The 5th
International Congress & Exhibition of the ISIS of Symmetry. Sydney, 2001. Intersections of Art and Science:
http://www.isis-s.unsw.edu.au/interact/gallery/image_files/zenkin/a_zenkin.html
Zenkin A.A. Cognitive Visualization of the Continuum Problem and of the Hyper-Real Numbers Theory. International Conference "Analyse et Logique", UMH, Mons, Belgia, 1997. Abstracts, pp. 93-94 (1997).
A.A.Zenkin, Does Axiomatic Set Theory Need New Axiom? - 12th. International Congress of Logic,
Methodology and Philosophy of Science (LMPS-2003). Oviedo (Spain), August 7-13, 2003 Abstract. (Has been
accepted for publication. See at:
http://webuniovi.innova.uniovi.es/Congresos/2003/DLMPS/Contributed/Accepted.htm)
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