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Modern system of mathematics and a pair of hidden contradictions in its foundation

Wujia Zhu (Department of Computer Science, Nanjing University of Aeronautics and Astronautics, Nanjing, People's Republic of China)
Yi Lin (College of Economics and Management, Nanjing University of Aeronautics and Astronautics, Nanjing, People's Republic of China Department of Mathematics, Slippery Rock University, Slippery Rock, Pennsylvania, USA)
Ningsheng Gong (College of Information Science, Nanjing University of Technology, Nanjing, People's Republic of China)
Guoping Du (Institute of Modern Logic and Applications, Nanjing University, Nanjing, People's Republic of China)

Kybernetes

ISSN: 0368-492X

Article publication date: 11 April 2008

Abstract

Purpose

The paper's aim is to show a pair of deeply hidden contradictions in the system of mathematics.

Design/methodology/approach

The paper takes a conceptual approach to the problem.

Findings

It is indicated that it is an intrinsic attribute of modern mathematics and its theoretical foundation to mix up the intensions and methods of two different thoughts of infinities, which provides the basis of legality for using the methods of analysis, produced by combining the two kinds of infinities, in the study of the modern mathematical system. In this paper, by exactly employing the method of analysis of mixing up potential and actual infinities, we card the logical and non‐logical axiomatic systems for modern mathematics. The outcome of our carding implies that in modern mathematics and its theoretical foundation, some axioms implicitly assume the convention that each potential infinity equals an actual infinity, while some other axioms implicitly apply the belief that “each potential infinity is different of any actual infinity.”

Originality/value

By using the concepts of potential and actual infinities, the authors uncover two contradictory thinking logics widely employed in the study of mathematics.

Keywords

Citation

Zhu, W., Lin, Y., Gong, N. and Du, G. (2008), "Modern system of mathematics and a pair of hidden contradictions in its foundation", Kybernetes, Vol. 37 No. 3/4, pp. 438-445. https://doi.org/10.1108/03684920810863390

Publisher

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Emerald Group Publishing Limited

Copyright © 2008, Emerald Group Publishing Limited