Geodesic
Scientific intuition of Genii against mytho-‘logic' of Cantor's transfinite ‘paradise'
Philosophia Scientiae, Aperçus philosophiques en logique et en mathématiques, Tome 9 (2005) no. 2, pp. 145-163

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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.

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     author = {Zenkin, Alexander A.},
     title = {Scientific intuition of {Genii} against mytho-{\textquoteleft}logic' of {Cantor's} transfinite {\textquoteleft}paradise'},
     journal = {Philosophia Scientiae},
     pages = {145--163},
     publisher = {\'Editions Kim\'e},
     volume = {9},
     number = {2},
     year = {2005},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/PHSC_2005__9_2_145_0/}
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Zenkin, Alexander A. Scientific intuition of Genii against mytho-‘logic' of Cantor's transfinite ‘paradise'. Philosophia Scientiae, Aperçus philosophiques en logique et en mathématiques, Tome 9 (2005) no. 2, pp. 145-163. http://geodesic.mathdoc.fr/item/PHSC_2005__9_2_145_0/