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:





[Plus d'informations -en français/in french-]


(CMAP28 WWW site: this page was created on 01/25/2019 and last updated on 10/08/2022 10:56:15 -CEST-)



[See all related pictures (including this one) [Voir toutes les images associées (incluant celle-ci)]]

[Please visit the related ImagesDidactiques picture gallery [Visitez la galerie d'images ImagesDidactiques associée]]
[Please visit the related NumberTheory picture gallery [Visitez la galerie d'images NumberTheory associée]]

[Go back to AVirtualMachineForExploringSpaceTimeAndBeyond [Retour à AVirtualMachineForExploringSpaceTimeAndBeyond]]

[The Y2K Bug [Le bug de l'an 2000]]

[Site Map, Help and Search [Plan du Site, Aide et Recherche]]
[Mail [Courrier]]
[About Pictures and Animations [A Propos des Images et des Animations]]


Copyright © Jean-François Colonna, 2019-2022.
Copyright © CMAP (Centre de Mathématiques APpliquées) UMR CNRS 7641 / Ecole Polytechnique, 2019-2022.