Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise'
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 ac...
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2005
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| Acesso em linha: | Accès Université d'Orléans et IFPM Accès Université d'Orléans et IFPM |
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cairn-SCIE_092_0010SCIEPhilosophia Scientiæ https://shs.cairn.info/revue-philosophia-scientiae-2005-2-page- ?lang=en https://doi.org/10.4000/philosophiascientiae.532 Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' Zenkin, Alexander A.2005 fre Philosophia Scientiæ | 9-2 | 2 | 2005-11-01 | p. - | 1281-2463 SCIE_0925 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.Cairn free access |
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Article ou chapitre numérique |
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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. |
| author |
Zenkin, Alexander A. |
| spellingShingle |
Zenkin, Alexander A. Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' |
| author_facet |
Zenkin, Alexander A. |
| author_sort |
Zenkin, Alexander A. |
| title |
Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' |
| title_short |
Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' |
| title_full |
Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' |
| title_fullStr |
Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' |
| title_full_unstemmed |
Scientific Intuition of Genii Against Mytho-‘Logic' of Cantor's Transfinite ‘Paradise' |
| title_sort |
scientific intuition of genii against mytho-‘logic' of cantor's transfinite ‘paradise' |
| publishDate |
2005 |
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| container_issue |
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| url |
https://ezproxy.univ-orleans.fr/login?url=https://shs.cairn.info/revue-philosophia-scientiae-2005-2-page-
?lang=en https://ezproxy.univ-orleans.fr/login?url=https://doi.org/10.4000/philosophiascientiae.532 |
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1877246818749775872 |