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1. INTRODUCTION.
COGNITIVE VISAULIZATION
OF NUMBER-THEORETICAL
ABSTRACTIONS

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 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 - 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) 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.



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