1. INTRODUCTION.
COGNITIVE VISUALIZATION
OF NUMBERTHEORETICAL
ABSTRACTIONS
ABSTRACT: As is known, G.Cantor formulated his famous Continuum Hypothesis in
the end of the XIX Century. Metamathematics 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 metamathematics 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 metamathematical achievements by Kurt Gödel
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 metamathematical 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 complete absence of any progress in the Continuum Hypothesis proof (or dispoof)
on the way of modern metamathematics 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 NONmetamathematical and NONmathematicallogic
way based on a socalled scientific cognitive computer visualization technique
 to a new comprehension of the Continuum Problem itself is offered below.
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 NTproblems (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 CVapproach, we visualized
this twice abstract connection in the form of colormusical 2Dimages
(socalled pythograms) of abstract NTobjects, and obtained really a
lot of new NTresults. In particularly, we generalized wellknown
Classical Waring's Problem, generalized and proved the famous theorems
of Hilbert, Lagrange, Wieferich, Balasubramanian, Desouillers, and
Dress, discovered a new type of NTobjects, a new universal additive
property of the natural numbers and a new method,  the socalled
SuperInduction method,  for the rigorous proving of general
mathematical statements of the form "n
P(n) with the help of CVImages, where P(n) is a NTpredicate.
By means of the CVapproach and the SuperInduction 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 NTtheorems was proved
rigorously in the framework of GWP [1 
4].
In this paper, we use ideas of this
CVapproach for the cognitive visualization of some basic number
systems in classical Set theory and NonStandard Analysis. We believe
that the essense of the classical (G.Cantor) Set Theory consists in the
Continuum Problem. 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 Hyperproofs , and prove some rather unusual settheoretical
statements basing on the CVimage of Continuum Problem. Finally, we produce
a new classification of number systems that clarify a particular role and
place of the hyperreal numbers system of nonstandart analysis in the
modern metamathematics [7, 10,
11]. Some unexpected but quite natural
connections between the CVimage of Continuum Problem and Leibniz's
Monadology ideas are presented.
