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COGNITIVE (SEMANTIC) COMPUTER VISUALIZATION
OF THE CONTINUUM PROBLEM
AND SOME MIRROR-SYMMETRIC PROOFS
IN THE G.CANTOR TRANSFINITE NUMBERS THEORY.
A.A.Zenkin
(Computer Center of the Russian Academy of Sciences, Moscow)
(e-mail: [email protected] )
Table of Content.
1. Introduction. Cognitive Visaulization Of Number-Theoretical Abstractions.
2. Different formulations of Continuum Hypothesis.
3. Different Representations Of The Real Numbers x  [ 0, 1 ].
4. Cognitive Visualization Of Continuum Problem.
5. New Kind Of G.Cantor's Transfinite Integers of the OMEGA type.
6. Some Mirror-Like Proofs Of Mirror-Like Theorems.
7. "Periodical System" of Hyper-Real Numbers.
8. Mathematical Sciences Classification Based On the "Periodical System" of Hyper-Real
Numbers
9. Main Conclusions. Cognitive Visualization Of Leibniz's Monadology.
10. References
1. INTRODUCTION. COGNITIVE (SEMANTIC) VISAULIZATION OF NUMBERTHEORETICAL ABSTRACTIONS.
ABSTRACT. - As is known, G.Cantor formulated his famous Continuum Hypothesis in the
end of the XIX Century. Meta-mathematics and mathematical logic appeared some decades
later, in the first half of the XX Century. So, that Continuum Hypothesis can not "genetically"
be a problem of and have an attitude to either modern meta-mathematics or modern
mathematical logic. Of course, it is not infrequently in the science when new methods of one
science area can help to solve quite old problems of another its area. For example, the famous
meta-mathematical achievements by Kurt Geodel and Paul J.Cohen helped to prove the
independence of Continuum Hypothesis in a framework of an axiomatic, say, ZermeloFrenkel's set theory. But even P.J.Cohen himself, - concerning the solvability of Continuum
Hypothesis by means of modern meta-mathematical methods, - wrote in his famous
monography [8]: "... Continuum Hypothesis is a rather dramatic example of what can be
called (from our today's point of view) an absolutely undecidable assertion, ..." (p.13). The
1
complete absence of any progress in the Continuum Hypothesis proof (or dispoof) on the way
of modern meta-mathematics during last decades confirms the validity of Cohen's pessimism.
So, it is obviously that new ways are necessary here. One of such new ways, - a NON-metamathematical and NON-mathematical-logic way based on a so-called scientific cognitive
computer visualization technique, - to a new comprehension of the Continuum Problem itself
is offered below.
So, Cognitive Visualization (CV) aims to represet an essense of a scientific abstract
problem domain, i.e. the most principal connections and relations between elements of that
domain, in a graphic form in order to see and discover an essentially new knowledge of a
conceptual kind [1]. For example, in classical Number Theory (NT) such the main feature
giving rise to many famous NT-problems (such as Fermat's, Goldbach's, Waring's problems)
is, by B.N.Delone and A.Ya.Hintchin, a hard comprehended connection between two main
properties of natural numbers - their additivity and multiplicativity. Nevertheless, by means of
CV-approach, we visualized this twice abstract connection in the form of color-musical 2Dimages (so-called pythograms) of abstract NT-objects, and obtained really a lot of new NTresults. In particularly, we generalized well-known Classical Waring's Problem, generalized
and proved the famous theorems by Hilbert, Lagrange, Wieferich, Balasubramanian,
Deshouillers, and Dress, discovered a new type of NT-objects, a new universal additive
property of the usual finite natural numbers and a new method, - the so-called SuperInduction method, - for the rigorous proving of general mathematical statements of the form
nP(n) with the help of CV-Images, where P(n) is a NT-predicate. By means of the CVapproach and the Super-Induction method, the Generalized Waring's Problem (GWP) was
seen (in direct sense of the word) and formulated. The complete solution of GWP was given
and a lot of fundamentally new NT-theorems was proved rigorously in the framework of
GWP [1-4].
In this paper, we use ideas of this CV-approach for the cognitive visualization of some
basic number systems in classical Set Theory and Non-Standard Analysis. We believe that the
essense of the classical (G.Cantor's) Set Theory consists in the Continuum Problem that is a
problem on a Nature of the connection between the deepest mathematical and philosophical
conceptions of the "discreteness" and the "continuity". Therefore, first of all, we visualize this
Problem. Then we use the J.Barwise, J.Etchemendy and E.Hammer [5, 6, 13] ideas on
Multumedial and Hyper-proofs , and prove some rather unusual set-theoretical statements
basing on the CV-image of Continuum Problem. Finally, we produce a new classification of
number systems that clarify a particular role and place of the hyper-real numbers system of
non-standart analysis in the modern meta-mathematics [7,10,11]. Some unexpected but quite
2
natural connections between the CV-image of Continuum Problem and Leibniz's Monadology
ideas are presented.
2. SOME FORMULATIONS OF CONTINUUM HYPOTHESIS.
Introduce the following notations.
For any set X, its power (cardinality) we shall denote by |X| or Card{X}.
Denote the set of all finite natural numbers by N = {1, 2, 3, ...}. Since N is a countable
set, then |N| = 0 .
Denote the set of all real numbers (of all proper fractions) of the segment [0,1] by D.
Since D has, by the well-known Cantor's theorem, the power C of Continuum, then |D| = C.
Now, there are two following main formulations of Continuum Hypothesis [8].
1) The classical Cantor Continuum Hypothesis formulation: C=1 .
2) The generalized Continuum Hypothesis formulation, by Cohen:  |P()| = +1., where
P() is the power-set of any set X with Card{X} =  .
As is known, P.J.Cohen completes his monography [8] by the following estimation of
the Continuum Cardinality: "Thus, C is greater than n ,  ,  , where  =  , and so on.
" (p.282). Therefore, we shall even not try to imagine visually a set of integers of a cardinality
succeeding 0, and use the following most weak formulation of Continuum Hypothesis.
3) Is there a set of integers, say M, such that a 1-1-correspondence between that set M and the
set D of all real numbers (proper fractions, points) of the segment [0,1] can be realized?
That is
Whether there exists even one such a set M of integers, that |M| = C ?
We shall indicate below just such the set M, construct such the 1-1-correspondence, and
prove that the set M has the continuum cardinality, C.
3
3. DIFFERENT REPRESENTATIONS OF THE REAL NUMBERS x  [ 0, 1 ].
There are the following main ways for representing and interpreting real numbers
xD = [0,1].
1. As an individual geometrical point x of the segment [0,1].
2. In an arbitrary, for example, binary positional number system:
as an infinite sequence of binary digits:
x = 0 . a1 a2 a3 ... an ..., where n [[an =0] or [an =1]].

a n  2  n (  1).
3. As the infinite sum: x =
n 1
4. As an infinite path on the binary Tree: V a1 a2 a3 ... an ... (Fig.1.)
The rather known cognitive image of the last representation of all real numbers x  D is
shown in Fig.1.

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]
Fig.1. Cognitive visual model of the real numbers set, D, on the segment, [0,1].
 A.Z., 1995.
4
4. COGNITIVE (SEMANTIC) VISUALIZATION OF CONTINUUM PROBLEM.
“Transfinite Numbers themselves are, in a certain sense,
new irrationalities. Indeed, in my opinion, the method
for the definition of finite irrational numbers is quite
analogous, I can say, is the same one as my method for
introducing transfinite numbers. It can be certainly
said: transfinite numbers stand and fall together with
finite irrational numbers.”
- Georg Cantor.
Denote the tree in Fig. 1 by TR , rotate Fig. 1 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-like 1-1correspodence, say , between the original tree TR and its mirror image - the tree TL . The
interpretation of the levels of the trees TR and TL is also given in Fig. 2.


x =
 a i  2i
x =
i0
a)  ... +1  ... n ... 3
b) 2 ...2+1 2 ... 2n ... 22
c) a  ... a +1 a  ... a n ... a 2
2
21
 a i  2-i
i 1
1
20
a1 a0



1
2
3 ... n ...  +1 ... 
-1
-2
2 2 2-3 ... 2-n ... 2- 2-(+1) ... 2-
a1 a2 a3 ... an ... a a+1 ... a
Fig.2. Cognitive Visualization of Continuum Problem and the hyper-real numbers system:
a) level 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., 1995.
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5. NEW KIND OF TRANSFINITE INTEGERS OF G.CANTOR'S -TYPE.
Consider a real number, x  [0,1] :

 x =  a i 2  i ,  i [[ai =0] or [ai =1]].
x = 0.a1 a2 a3 ... ai ...
(1)
i=1
There are only two following cases of interest.
CASE 1. x is a rational number with a 0-tail, i.e.  k  i > k [ai =0].
By virtue of , this x is corresponded by an integer x possessing the following property:

k

x  x =...a k ... a 2 a1 a 0   a i 2 i   a i 2 i   , so x is a finite natural number.
i=0
i=0
Q0 is the set of all rational fractions x  [0,1] with the 0-tail;
N is the set of all finite natural numbers.
Thus, we have visually proved
THEOREM 1. Q0 is -equivalent to N, i.e. Card {Q0} = Card {N} = 0 .
Designate:
CASE 2. x is an irrational number, i.e.  k  i > k [ai = 1].
By virtue of , this x is corresponded by an integer x possessing the following property:

x  x =...a i ... a 2 a1a 0 =

 a i 2i ,
(2)
i=0
so we have, generally speaking, a new mathematical object (2), with the same ontological
status as the usual irrational number (1). It is quite appropriate here, to remind of the wellknown G.Cantor's words: "...the transfinite numbers exist in the same sense as the finite
irrational numbers."
Consider the main properties of this new mathematical objects.
THEOREM 2. For all x : Ord { x } = .
PROOF.
1.1. x is a transfinite integer, since it is a countable sum of finite natural numbers, and
consequently nN[ x > n].
1.2. x   , since  is the least transfinite integer, by G.Cantor's definition.

2.1. x =
2
i=0
i=0

2.2.

 ai 2i 
i
,
since, in general case, not all a i in x are equal to zero.
k
 2 i = lim  2 i , by the classical mathematical analysis definition of the -sum
k 
i=0
i=0
limit.
k
k
i=0
i=0
2.3. lim  2 i = lim { 21, 22, ... , 2k, ... } , since k1 [  2 i  2 k 1 ].
k 
2.4. lim { 21 , 22, ... , 2k, ... }  lim { 1, 2, ... , k, ... } =  , since the first is a subsequence
of the second, and by Cantor's (and, for example, by F.Hausdorff's [9] ) definition of the least
transfinite integer .
2.5. x   .
3.1. From 1.2 and 2.5, we have   x

and consequently Ord { x } = .
The Theorem is proved.
6
x
Denote the set of all transfinite integers x as
. Then we get
Corollary 1. For any x 
: Card( x ) = 0 .
x
Further, by , we have the following property of all x 
Property 1. For any two real numbers x1  D and x2  D,
x.

(x1  x 2 )  (x1  x2 ) ,
that is any two transfinite integers x 1 and x 2 are different mathematical objects in the same
sense as any two real numbers x1 and x2 are different in a sense of Classical Mathematics.
x
From this Property 1 and the definition of an element x  , we have
Corollary 2. Every infinite subsequence of the series of natural numbers is an
individual mathematical object representing a transfinite ordinal -type number.
Further, denote the set of all irrational numbers of the segment [0,1] by Dir. Then, by
virtue of Theorem 2 and Property 1, we have
THEOREM 3: Dir is -equivalent to
x , i.e. Card { x } = Card {Dir} = C.
Thus, this Theorem solves the Continuum Problem in its weak formulation 3 (see
above).
Further, we have
x
Property 2. For any n-level of the tree TL , all the transfinite integers x 
with
confinal heads (from the n-level to the -level of TL ) form a finite set, say Gn , consisting of
2n transfinite integers. Thus, Gn is a "Galaxy" [ 10,11 ] since the difference between any two
transfinite integers of Gn is a finite integer.
Lastly, the mirror mapping  possesses the following interesting but rather unique
Property 3. The mirror mapping  transforms the ordered continual set D (without the
countable set of all rational fractions with 0-tails) into the unordered set
of all transfinite
integers x of the -type.
Remark, however, that the set
can be ordered (not well-ordered !) by means of the
natural order given on D: for example, if x1 < x2 then x 1 < x 2.
x
x
7
6. SOME MIRROR-LIKE PROOFS OF SOME MIRROR-LIKE THEOREMS.
Probably, the first use of mirror-arguments in meta-mathematical proofs belong to
S.Kleene (see his proof of the Cantor-Bernstein Theorem in [12], p. 18 ). Modern and much
more advanced and systematic investigations in this very promising area of logic are
elaborating in Visual Inference Lab of the Indiana University by J.Barwise, J. Etchemendy,
E.Hammer [13]
Using their ideas on symmetry and multimedia argumentation (the visual proof, almost
by L.E.J.Brouwer) and results obtained above, we formulate some mirror-statements (mirrorlike Theorems) based on the cognitive visual image of the 1-1-correspondence,  , between
the binary trees TR and TL shown in Fig. 2.
M-THEOREM 1. IF the geometrical point x of the segment [0,1] is an individual object
THEN the corresponding infinite path x of the tree TR attaines its -level.
COROLLARY 1. All infinite paths x of the tree TR attaine its -level.
COROLLARY 2. The path x = 0.000...1 (the geometric poit in usual sense) is the maximal
transfinite small number (maximal infinitesimal): since |x| = 2 -  < 1/n by any finite natural n.
M-THEOREM 2. IF an infinite path x of the tree TR attaines the -level THEN the
corresponding infinite path x of the tree TL attaines the -level of the tree TL.
COROLLARY 1. All infinite paths x of the tree TL attaine its -level.
M-THEOREM 3. IF a path x of the tree TL attaines the -level THEN Ord{ x }=.
COROLLARY 1. The path x = 1 ... 000. is the minimal transfinite large number:
since | x | = Card { x } = 2  = 0 .
COROLLARY 2 By virtue of , Card{all x TL } = Card{all xTR }.
M-THEOREM 4. IF the geometrical point x of the segment [0,1] exists as an individual
object THEN there exists the Cantor least trans-finite integer .
M-THEOREM 5. IF a path x of the tree TL attaines the -level THEN there exist (by
Peano!) (+1)-th level in the tree TL and any infinte path x attaines the (+1)-level.
COROLLARY 1. All infinite paths x of the tree TL attaine its (+1)-level.
M-THEOREM 6. IF the transfinite (+1)-level in the tree TL exists THEN there exists the
corresponding transfinite (+1)-level of the tree TR.
COROLLARY 1. All transfinite paths x of the tree TR attaine its (+1)-level.
COROLLARY 2. There is the infinitesimal x = 0.000...0.1+1. of the order (+1) and
Card{x}=2-(+1)
COROLLARY 3. There exist infinitesimals x of any transfinite-small order , so that
Ord{ x } =  and Card { x } = 2-.
Remark here that all the Mirror-Theorems are condition statements. Therefore all
objections and doubts concerning the tree TL (particularly, the existence and rather unusual
properties of the transfinite integers x  ) are mirror-likely reflected into the same
objections and doubts concerning the tree TR , in particular, the existence and properties of the
usual real numbers xD, or, that is the same, of the usual geometric points of the segment
[0,1].
Some obvious methodological consequences of the consideration above are presented in
Fig. 3 and Fig. 4.
x
8
7. "PERIODICAL SYSTEM" OF HYPER-REAL NUMBERS.
v
Levels:.  ... +1  ... n ... 3 2 1  1 2 3 ... n ...  +1 ...  ...
v
FINITE STANDARD REAL NUMBERS (EXCLUDING BOTH - Levels)
v
NON-STANDARD PROPER FRACTIONS (INCLUDING THE RIGHT - Level)
v
NON-STANDARD HYPER-NATURAL NUMBERS (INCLUDING THE LEFT - Level)
OR CANTOR'S TRANSFINITE INTEGERS OF THE -ORDER.
v
THE SYSTEM OF NON-STANDARD HYPER-REAL, "IN-BOTH-SIDES" TRANSFINITE
NUMBERS AS A WHOLE.
Fig. 3. "Periodical system" of hyper-real numbers.
8. MATHEMATICAL SCIENCES CLASSIFICATION BASED ON THE
"PERIODICAL SYSTEM" OF HYPER-REAL NUMBERS
v
Levels:.  ... +1  ... n ... 3 2 1  1 2 3 ... n ...  +1 ...  ...
v
COMPUTER MATHEMATICS (EXCLUDING BOTH - Levels)
v
CLASSICAL MATHEMATICAL ANALYSIS (INCLUDING BOTH - Levels)
(THE POINTS AT THE BOTH - Levels ARE IDEAL ELEMENTS,
IN LEIBNIZ'S AND HILBERT'S SENSE)
v
CANTOR'S THEORY OF TRANSFINITE INTEGERS
v
NON-STANDARD ANALYSIS AS A WHOLE
Fig. 4. Mathematical sciences classification
based on the "periodical system" of hyper-real numbers
 A.Z., 1995.
9
9. MAIN CONCLUSIONS. COGNITIVE VISUALIZATION OF LEIBNIZ'S
MONADOLOGY.
1. By means of the Cognitive ( Semantic ) Computer Visualization approach, a new
class of transfinite integers of G'Cantor's -type is discovered. Call here that class
2. The natural mirror-like 1-1-correspondence between the set
numbers (points) of the segment [0,1] is established, so that the set
x.
x and the set of all real
x of the -type integers
has the continuum cardinality.
3. The informal (and, consequently, formal) incompleteness of Cantor's transfinite
integers Theory (and, consequently, of all modern axiomatic set theories) is proved.
4. It is shown that the First Hilbert Problem - the Continuum Problem - is a PSEUDOproblem of "naive" G.Cantor's (and any modern axiomatic) Set Theory.
5. The intrinsic natural extension of classical Cantor's Theory of transfinite -type
integers is constructed and a natural solution of Continuum Problem, in its weak formulation
3, is offered in the framework of this extension.
6. The natural cognitive visual representation of the hyper-real numbers system and the
corresponding visual interpretation of the modern non-standard analysis as a whole are
proposed.
7. The natural cognitive visualization of Leibniz's Monadology is realized (See Fig. 2):
every vertex xTR , i.e., a simple Monad in Leibniz's sense, is a root of the same T R ,
(outgoing to the right), i.e., an original Universe, in Leibniz's sense. The same refers to the
mirror-symmetrical tree TL . Now, look narrowly, please, at the Fig. 2. And every VISUAL
THINKING MIND'S EYE will see and hear the following thoughts expressed many years ago
[14] by the Great Philosopher and Mathematician,
GOTTFRIED WILHELM LEIBNIZ:
9. Indeed, each Monad must be different from every other… These Monads are the real
atoms of nature and, in a word, the elements of things.
11. It follows from what has just been said, that the natural changes of the Monads come
from an internal principle, since an external cause can have no influence upon their inner
being. (Theod. 396, 400.)
12. But, besides the principle of the change, there must be a particular series of changes [un
detail de ce qui change], which constitutes, so to speak, the specific nature and variety of the
simple substances.
13. This particular series of changes should involve a multiplicity in the unit [unite] or in
that which is simple. For, as every natural change takes place gradually, something changes
and something remains unchanged; and consequently a simple substance must be affected and
related in many ways, although it has no parts.
16. We have in ourselves experience of a multiplicity in simple substance, when we find that
the least thought of which we are conscious involves variety in its object. Thus all those who
admit that the soul is a simple substance should admit this multiplicity in the Monad;
10
51. But in simple substances the influence of one Monad upon another is only ideal, and it
can have its effect only through the mediation of God, in so far as in the ideas of God any
Monad rightly claims that God, in regulating the others from the beginning of things, should
have regard to it. For since one created Monad cannot have any physical influence upon the
inner being of another, it is only by this means that the one can be dependent upon the other.
(Theod. 9, 54, 65, 66, 201. Abrege, Object. 3.)
53. Now, as in the Ideas of God there is an infinite number of possible universes, and as only
one of them can be actual, there must be a sufficient reason for the choice of God, which leads
Him to decide upon one rather than another. (Theod. 8, 10, 44, 173, 196 sqq., 225, 414-416.)
56. Now this connexion or adaptation of all created things to each and of each to all, means
that each simple substance has relations which express all the others, and, consequently, that
it is a perpetual living mirror of the universe. (Theod. 130, 360.)
57. And as the same town, looked at from various sides, appears quite different and becomes
as it were numerous in aspects [perspectivement]; even so, as a result of the infinite number
of simple substances, it is as if there were so many different universes, which, nevertheless
are nothing but aspects [perspectives] of a single universe, according to the special point of
view of each Monad. (Theod. 147.)
58. And by this means there is obtained as great variety as possible, along with the greatest
possible order; that is to say, it is the way to get as much perfection as possible. (Theod. 120,
124, 241 sqq., 214, 243, 275.)
62. Thus, although each created Monad represents the whole universe, it represents more
distinctly the body which specially pertains to it, and of which it is the entelechy; and as this
body expresses the whole universe through the connexion of all matter in the plenum, the soul
also represents the whole universe in representing this body, which belongs to it in a special
way. (Theod. 400.)
63. The body belonging to a Monad (which is its entelechy or its soul) constitutes along with
the entelechy what may be called a living being, and along with the soul what is called an
animal. Now this body of living being or of an animal is always organic; for, as every Monad
is, in its own way, a mirror of the universe, and as the universe is ruled according to a perfect
order, there must also be order in that which represents it, i.e. in the perceptions of the soul,
and consequently there must be order in the body, through which the universe is represented
in the soul. (Theod. 403.)
Acknowledgements. - The author wish to thank the International Science Foundation of
G.Soros (Grant No. ZZ5000/114, 1995), the Russian Humanitarian Scientific Foundation
(Grant No. 98-03-04348), the Russian Foundation for Basic Researches (Grant No. 98-0100339) and Ministry of Science and Technologies of Russian (Grants No. 05.04.1179, No.
05.04.1221, 1996-1997) for the financial support of this work. Also, the author wish to thank
his permanent co-author and collaborator, system programmer, Anton Zenkin,
11
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Translated from Doklady Alademii Nauk, Vol 354, No. 5, 1997, pp. 587 - 589.
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Cambridge University Press, 1994.
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Computation, Stanford: CSLI and FoLLI, 1995.
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Numbers Theory. - International Conference "Analyse et Logique", UMH, Mons, Belgia, 2529 August, 1997. Abstarcts, pp. 93-94 (1997).
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8..Cohen, Paul J. Set Theory and the Continuum Hypothesis. - Moscow : MIR, 1969.
9. Hausdorff, F. Set theory. - Moscow - Leningrad : GINTL, 1937, p.70.
10.Davis M. Applied nonstandard analysis. - JOHN WILLEY & SONS, N.Y., 1977.
11. Uspensky, V.A. What is nonstandard analysis like? - Moscow : NAUKA, 1987.
12. S.C.Kleene, Introduction to Metamathematics. - D.Van Nostrand Company, Inc., N.Y.
- Toronto, 1952.
13. Hammer, Eric M. Symmetry as a method of proof. J. Philos. Logic, v.25 (1996), no.5,
523-543.
14. LEIBNIZ Monadology 1898; THE MONADOLOGY by Gottfried Wilhelm Leibniz
translated by Robert Latta;
http://english-www.hss.cmu.edu/philosophy/leibniz-monadology.txt
Correspondence address: Prof. A.A.Zenkin,
Doctor of Phys.& Math. Sciences,
Leading Research Scientist of Computing Center of the RAS,
Member of the Russian Philosophical Society and AI-Association
Full-Member of the International Federation of Artists.
Telephone: (7-095) 337.2168
E-mail: [email protected]
WEB-Site: http://www.com2com.ru/alexzen
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