9. MAIN CONCLUSIONS.
OF LEIBNIZ'S MONADOLOGY
1. By means of the Cognitive (Semantic) Computer Visualization approach, a new class of transfinite integers of G'
Cantor's w-type is discovered. Call here that class X-.
GOTTFRIED WILHELM LIEBNIZ:
2. The natural mirror-like 1-1-correspondence between the set X- and the set of all real numbers (points) of the segment [0,1] is established, so that the set X- of the w-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 PSEUDO-problem of "naive" G.Cantor's (and any modern axiomatic) Set Theory.
5. The intrinsic natural extension of classical Cantor's Theory of transfinite w-type integers is constructed and a natural solution of Continuum Problem, in its weak formulation  , 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 xÎ TR,
i.e. a simple Monad in Leibniz's sense, is a root of the same TR
(outgoing to the right), i.e., an original Universe, in Leibniz's sense. The same refers to the mirror-symmetrical tree T
L. 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
 by the Great Philosopher and Mathematician,
9. Indeed, each Monad must be different from every other…
These Monads are the real atoms of nature and, in a word, the elements
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,
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
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,
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.,
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
Basic Researches Foundation (Grant No. 98-01-00339) 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.