Navigation – Plan du site

AccueilNuméros9-2Scientific Intuition of Genii Aga...

Scientific Intuition of Genii Against Mytho-‘Logic’ of Cantor’s Transfinite ‘Paradise’

Alexander A. Zenkin
p. 145-163

Résumé

In the paper, a detailed analysis of some new logical aspects of Cantor’s diagonal proof of the uncountability of continuum is presented. For the first time, strict formal, axiomatic, and algorithmic definitions of the notions of potential and actual infinities are presented. It is shown that the actualization of infinite sets and sequences used in Cantor’s proof is a necessary, but hidden, condition of the proof. The explication of the necessary condition and its factual usage within the framework of Cantor’s proof makes , Cantor’s proof invalid. It’s shown that traditional Cantor’s proof has a second necessary, but hidden as well, condition which is teleological by its nature, i.e., is not mathematical. The explication of the second necessary condition makes Cantor’s statement on the uncountability of continuum unprovable from the point of view of classical logic.
One of the most dramatic facts in history of science is connected just with the notion of actual infinity and consists in the following. On the one hand, Aristotle, Berkeley, Locke, Descartes, Spinoza, Gauss, Kant, Cauchy, Kronecker, Hermite, Poincaré, Bair, Borel, Brouwer, Quine, Wittgenstein, Weyl, Luzin, and a lot of other outstanding creators of classical logic and classical mathematics, during millenniums, stated categorically and insistently warned about that the actual infinity is a self-contradictory notion and its usage in mathematics is inadmissible (e.g., according to Poincare, “There is no actual infinity. - Cantorians forgot that and fell into contradictions. Later generations will regard Mengenlehre (set theory) as a disease from which one has recovered”). On the other hand, a lot of outstanding scientists of the XX c.c., such as Hilbert, Church, Turing, Gödel, etc., and modern axiomatic set theory as a whole ignored this (intuitive though) opinion of genii and accepted Cantor’s “transfinite paradise”, based on the actual infinity. In the paper some logical, philosophical, and psychological aspects of this historical “favorable opposition” are analyzed, and the fact that the scientific intuition of genii has won the historical battle against the mytho-’logic’ of Cantor’s transfinite ‘paradise’ is explained and logically justified.

Haut de page

Pour citer cet article

Référence papier

Alexander A. Zenkin, « Scientific Intuition of Genii Against Mytho-‘Logic’ of Cantor’s Transfinite ‘Paradise’ »Philosophia Scientiæ, 9-2 | 2005, 145-163.

Référence électronique

Alexander A. Zenkin, « Scientific Intuition of Genii Against Mytho-‘Logic’ of Cantor’s Transfinite ‘Paradise’ »Philosophia Scientiæ [En ligne], 9-2 | 2005, mis en ligne le 15 juin 2011, consulté le 28 mars 2024. URL : http://journals.openedition.org/philosophiascientiae/532 ; DOI : https://doi.org/10.4000/philosophiascientiae.532

Haut de page

Auteur

Alexander A. Zenkin

Computing Center after A.A.Dorodnitsyn of the Russian Academy of Science (Moscou)

Haut de page

Droits d’auteur

Le texte et les autres éléments (illustrations, fichiers annexes importés), sont « Tous droits réservés », sauf mention contraire.

Haut de page
Rechercher dans OpenEdition Search

Vous allez être redirigé vers OpenEdition Search