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 Gödel
and Paul J.Cohen helped to prove the independence of Continuum Hypothesis
in a framework of an axiomatic, say, Zermelo-Frenkel'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 :
"... 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 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-meta-mathematical 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.
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 . 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 2D-images
(so-called pythograms) of abstract NT-objects, and obtained really a
lot of new NT-results. In particularly, we generalized well-known
Classical Waring's Problem, generalized and proved the famous theorems
of Hilbert, Lagrange, Wieferich, Balasubramanian, Desouillers, and
Dress, discovered a new type of NT-objects, a new universal additive
property of the natural numbers and a new method, - the so-called
Super-Induction method, - for the rigorous proving of general
mathematical statements of the form "n
P(n) with the help of CV-Images, where P(n) is a NT-predicate.
By means of the CV-approach 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 -
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) 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 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 metamathematics [7, 10,
11]. Some unexpected but quite natural
connections between the CV-image of Continuum Problem and Leibniz's
Monadology ideas are presented.