Scientific intuition of Genii against mytho-‘logic’ of Cantor’s transfinite ‘paradise’

Alexander A. Zenkin

Philosophia Scientiae (2005)

  • Volume: 9, Issue: 2, page 145-163
  • ISSN: 1281-2463

Abstract

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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.

How to cite

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Zenkin, Alexander A.. "Scientific intuition of Genii against mytho-‘logic’ of Cantor’s transfinite ‘paradise’." Philosophia Scientiae 9.2 (2005): 145-163. <http://eudml.org/doc/103747>.

@article{Zenkin2005,
abstract = {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.},
author = {Zenkin, Alexander A.},
journal = {Philosophia Scientiae},
language = {eng},
number = {2},
pages = {145-163},
publisher = {Éditions Kimé},
title = {Scientific intuition of Genii against mytho-‘logic’ of Cantor’s transfinite ‘paradise’},
url = {http://eudml.org/doc/103747},
volume = {9},
year = {2005},
}

TY - JOUR
AU - Zenkin, Alexander A.
TI - Scientific intuition of Genii against mytho-‘logic’ of Cantor’s transfinite ‘paradise’
JO - Philosophia Scientiae
PY - 2005
PB - Éditions Kimé
VL - 9
IS - 2
SP - 145
EP - 163
AB - 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.
LA - eng
UR - http://eudml.org/doc/103747
ER -

References

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  17. [17] Zenkin, A.A.2004.— On Logic of “plausible” meta-mathematical misunderstandings, (All-Russian Conference “Scientific Session of MEPI-2004”.) Moscow Engineering & Physical Institute, Proceedings, 2004, 182-183. 
  18. [18] Zenkin, A.A.2003.— A priori logical assertions with a zero ontology, in Mathematics and Experience, A. G. Barabashev, (ed.), Moscow: Moscow State University Publishing House, 2003, 423–434. 
  19. [19] Zenkin, A.A.2002a.— As to strict definitions of potential and actual infinities, FOM-archive at: http://www.cs.nyu.edu/pipermail/fom/2002-December/006072.html 
  20. [20] Zenkin, A.A.2002b.— Gödel’s numbering of multi-modal texts, The Bulletin of Symbolic Logic, 8, (1), 180, 2002. 
  21. [21] Zenkin, A.A.2001.— Infinitum Actu Non Datur, Voprosy Filosofii (Problems of Philosophy), 9, 2001, 157–169. 
  22. [22] Zenkin, A.A.2000a.— Scientific Counter-Revolution in Mathematics, Moscow: NG-SCIENCE, Supplement to the Independent Newspaper: Nezavisimaya Gazeta, 19 July 2000, 13. 
  23. [23] Zenkin, A.A.2000b.— Georg Cantor’s Mistake, Voprosy Filosofii (Problems of Philosophy), 2, 2000, 165–168. See also: http://www.ccas.ru/alexzen/papers/The_Cantor_Paradise.htm, http://www.hist-analytic.org and a discussion about Cantor’s Diagonal Proof at: http://www.philosophy.ru/library/math/cantor.htm (in Russian). 
  24. [24] Zenkin, A.A.1999.— Cognitive (Semantic) Visualization of The Continuum Problem and Mirror Symmetric Proofs in the Transfinite Numbers Theory, Visual Mathematics, 1 (2). http://members.tripod.com/vismath1/zen/index.html. Zbl1044.03531MR1729760
  25. [25] Zenkin, A.A.1997a.— Cognitive Visualization of the Continuum Problem and of the Hyper-Real Numbers Theory, Intern. Conference “Analyse et Logique”, UMH, Mons, Belgia, 1997. Abstracts, 93–94. 
  26. [26] Zenkin, A.A.1997b.— “Cognitive Visualization of Some Transfinite Objects of Cantor Set Theory”, in Infinity in Mathematics: Philosophical and Historical Aspects, A. G. Barabashev, ed., Moscow: Janus-K, 1997, 77–91, 92–96, 184–189, 221–224. 

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