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4. COGNITIVE VISUALIZATION
OF CONTINUUM PROBLEM

"Transfinite Numbers themselves are, in a certain sense, new irrationalities. Indeed, in my opinion, the method for the definition of finite irrational numbers is quite analogous, I can say, is the same one as my method for introducing transfinite numbers. It can be certainly said: transfinite numbers stand and fall together with finite irrational numbers."

- Georg Cantor.

Denote the tree in Fig. 1 by TR, rotate the tree TR counter-clockwise at 90° and place a mirror AB at its root V in parallel to its levels. The visual result of such transformation we can see in Fig. 2, that demonstrates a Cognitive-Visual image of the mirror-like 1-1-correspodence, say Y, between the initial tree TR and its mirror image - the tree TL. The interpretation of the levels of the trees TR and TL is also given in Fig. 2.


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Fig. 2. Cognitive Visualization of Continuum Problem and the hyper-real numbers system:

a) level numbers of the trees;

b) powers of the base 2 in the binary system;

c) binary representation of the "in-both-side transfinite" hyper-real numbers.



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