JAMC
Identifying Cantor’s Diagonal Argument as an Antinomy: Exploring Complementary Analysis Techniques
TOTAL VIEWS: 441
Abstract
It has been noted that self-referential and ambiguous definition formulas are accompanied by complementary self-referential antinomy formulas, which give rise to contradictions. This made it possible to reexamine the ancient antinomies, Cantor’s Diagonal Argument (CDA), and the method of nested intervals, which is the basis for evaluating the existence of uncountable sets. CDA is seen by many mathematicians as a beautiful and easy argument whose consequences lead to different powers of infinity, thus opening the door to mathematical paradise. The simple reasoning he uses seems to be a highly effective way of defining a sequence, implying a proof of uncountability because the diagonals of a two-character list turn into opposites, and at first glance, there is nothing to disprove the argument. A new look at the complementarity of Cantor’s formulas in this article refreshes the hunches of Wittgenstein and other opponents of the existence of uncountable sets, putting CDA in a different light. In Cantor’s theorem, a formula was used to define a set that cannot be the value of any argument of any function f: ℕ → P(ℕ). Examining the complement of the created set, we find that this complement must be unique due to the bijective reversal 0 ↔ 1 of the signs of the indicator function of the Cantor set. However, at the same time, its definition generates two different sets for one argument, which contradicts the basic property of every function. Other studies confirm the invalidity of Cantor’s proofs and the nonexistence of uncountable sets.
References
[1] Shand AF. The Antinomy of Thought. Mind. 1890;15(59):357-72. Available from: http://www.jstor.org/stable/2247263. Accessed 24 Nov 2024.
[2] Tworak Z. Self-Reference and the Problem of Antinomies. Philosophical Issues in Science. 2008;16(2):43-58.
[3] Eldridge-Smith P. Paradoxes and hypodoxes of time travel. In: Jones JL, Campbell P, Wylie P, editors. Art and time. Melbourne: Australian Scholarly Publishing; 2007. p. 172-89.
[4] Billon A. Paradoxical Hypodoxes. Synthese. 2018. https://doi.org/10.1007/s11229-018-1711-1
[5] Richard J. Les principes des mathématiques et le problème des Ensembles. Revue Générale des Sciences Pures et Appliquées. 1905;16(541):142-4.
[6] Cantor G. Ueber eine Eigenschaft des Inbegriffes aller reellen algebraischen Zahlen. J Reine Angew Math. 1874;(77):258-62. doi:10.1515/crll.1874.77.258
[7] Cantor G. Über eine elementare Frage der Mannigfaltigkeitslehre. Jahresber Dtsch Math-Ver. 1891;1:75-8.
doi:10.1515/crll.1874.77.258
[8] Benacerraf P, Putnam H, editors. Philosophy of Mathematics. Prentice-Hall; 1964.
[9] Wittgenstein L. Zettel. Edited by von Wright GH, Translated by Anscombe GEM. University of California Press; 1970.
[10] Zhuang C. Wittgenstein’s analysis on Cantor’s diagonal argument. 2010. Available from: https://philarchive.org/rec/ZHUWAO
[11] Erdinç S. Contra Cantor: How to Count the “Uncountably Infinite”. Available from:
https://www.academia.edu/37229455/CONTRA_CANTOR_HOW_TO_COUNT_THE_UNCOUNTABLY_INFINITE_E_
[12] Good IJ. A note on Richard’s paradox. Mind. 1966;LXXV(299):431.
doi:10.1093/mind/LXXV.299.431
[13] Zhuang C. Wittgenstein’s analysis on Cantor’s diagonal argument. 2024.
[14] Molyneux P. Some critical notes on the Cantor Diagonal Argument. Open Journal of Philosophy. 2022;12(3):255-265.
[15] Padula J. Using historical proof-by-contradiction examples in senior mathematics: How Georg Cantor’s diagonal method made Alan Turing’s (1937-8) proof possible. Australian Mathematics Education Journal. 2023;5(3):37-41.
How to cite this paper
Identifying Cantor’s Diagonal Argument as an Antinomy: Exploring Complementary Analysis Techniques
How to cite this paper: Andrzej Burkiet. (2025) Identifying Cantor’s Diagonal Argument as an Antinomy: Exploring Complementary Analysis Techniques. Journal of Applied Mathematics and Computation, 9(2), 84-108.
DOI: http://dx.doi.org/10.26855/jamc.2025.06.001
More Articles
- Identifying Cantor’s Diagonal Argument as an Antinomy: Exploring Complementary Analysis Techniques
- Laplacian Matrices in Computer Graphics
- The Signless Laplacian Spectral Radius and Rainbow Matchings (Hamilton Paths) of Graphs
- Computing the Inversion of the Finite Hilbert Transform with Spline Wavelets
- The Cohesive Degree Condition of Non-separating Subtree Problems in Graph Partition
- A Neural Network-based Stock Timing Model