The Continuum Hypothesis (CH) [*L'Hypothèse du Continu (HC)*].

The so-called "power set" S(i+1)=P(S(i)) is the set of all subsets of S(i) (including the empty set as well as E itself).

This picture displays the first infinite sets S(i) by means of an "infinite ladder".
The bottom line exhibits:
S(0) = N (the integer numbers)

and the next one:
S(1) = R (the real numbers) **IF AND ONLY IF the Continuum Hypothesis -CH- is TRUE**.

The cardinal (ie. the number of elements) of S(i+1) is given by the following formula (Georg Cantor ~1890):
cardinal(S(i))
cardinal(S(i+1)) = 2

with:
cardinal(S(0)) = Aleph-zero (the first infinite cardinal)

And then the infinite sets S(i) are bigger and bigger when going up this ladder...

Obviously, one cannot display an infinite "object". In this picture each infinite set S(i) is exhibited as a finite
object made of a finite number of spheres. But this number increases strictly and in a non linear manner when going from S(i) to S(i+1).
And it is so in order to recall that a bijection between S(i) and S(i+1)
does not exist.

See some related pictures:

[pour plus d'informations -en français/in french-]

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Copyright (c) Jean-François Colonna, 2019.

Copyright (c) CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2019.