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1. INTRODUCTION.
COGNITIVE VISUALIZATION
OF NUMBER-THEORETICAL
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 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 [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 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 [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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