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11 – 20 of 20Wujia Zhu, Yi Lin, Guoping Du and Ningsheng Gong
This paper is the fourth part of the effort to resolve the following two problems that urgently need an answer: how can an appropriate theoretical foundation be chosen for modern…
Abstract
Purpose
This paper is the fourth part of the effort to resolve the following two problems that urgently need an answer: how can an appropriate theoretical foundation be chosen for modern mathematics and computer science? And, under what interpretations can modern mathematics and the theory of computer science be kept as completely as possible?
Design/methodology/approach
The paper is a conceptual discussion.
Findings
The paper lays out the set theoretical foundation for the mathematical system of potential infinities.
Originality/value
This work is the non‐logical axiomatic part of the mathematical system of potential infinities: the axiomatic set theoretic system. At the end, the problem of consistency of this axiomatic set theory is discussed.
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Wujia Zhu, Yi Lin, Ningsheng Gong and Guoping Du
The paper's purpose is to show a class of well‐used methods in set theory is invalid.
Abstract
Purpose
The paper's purpose is to show a class of well‐used methods in set theory is invalid.
Design/methodology/approach
A conceptual approach is taken.
Findings
Based on the concept of Cauchy theater introduced elsewhere, it is proven that the argument employing the diagonal method for the fact that the set of all real numbers R={x|r(x)}, where r(x)=df “x is a real number” has an uncountable cardinality, is not valid. The diagonal method has appeared in naive and modern axiomatic set theories.
Originality/value
Point out more paradoxes in the methodology of mathematics.
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Wujia Zhu, Yi Lin, Ningsheng Gong and Guoping Du
The paper's purpose is to analyze the concepts of potential and actual infinities.
Abstract
Purpose
The paper's purpose is to analyze the concepts of potential and actual infinities.
Design/methodology/approach
The exploration and research on potential and actual infinities generally touch on many disciplines, such as philosophy, logic, computer science, mathematics, etc. From the angle of a brief history, recall and development, the authors analyze the concepts of potential and actual infinities on one starting point and two locations to cut in.
Findings
Clarify the difference and connection of these two infinities on the level of mathematics and introduce the symbolized, descriptive definitions for potential and actual infinities.
Originality/value
It is the first time that the difference between the concepts of potential and actual infinities are clarified, which leads to the discovery of the fourth crisis in the foundations of mathematics.
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Wujia Zhu, Yi Lin, Guoping Du and Ningsheng Gong
The paper aims to show countable infinite sets are self‐contradictory non‐sets.
Abstract
Purpose
The paper aims to show countable infinite sets are self‐contradictory non‐sets.
Design/methodology/approach
The paper is a conceptual discussion.
Findings
Since, long ago, it has been commonly believed that the establishment and development of modern axiomatic set theory have provided a method to explain Russell's paradox. On the other hand, even though it has not been proven theoretically that there will not appear new paradoxes in modern axiomatic set theory, it has been indeed a century that no one has found a new paradox in modern axiomatic set theory. However, when we revisit some well‐known results and problems under the thinking logic of allowing two kinds of infinities, we discover that various countable infinite sets, widely studied and employed in modern axiomatic set theory, are all specious non‐sets.
Originality/value
A well‐known concept is shown to be not as correct as what has been believed.
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Wujia Zhu, Yi Lin, Ningsheng Gong and Guoping Du
The paper's aim is to show a pair of deeply hidden contradictions in the system of mathematics.
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.
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Wujia Zhu, Yi Lin, Ningsheng Gong and Guoping Du
The paper aims to introduce the concepts of potential and actual infinities.
Abstract
Purpose
The paper aims to introduce the concepts of potential and actual infinities.
Design/methodology/approach
A conceptual approach is taken.
Findings
It is a common belief that both Cantor and Zermelo completely employed the thinking logic of actual infinities in the naive and modern axiomatic set theory, and that Cauchy and Weierstrass completely applied that of potential infinities in the theory of limits. However, when it explores in depth the essential intensions of both potential and actual infinities, and after sufficiently understanding the difference and connections between the infinities and revisiting the realistic situations on how the concept of infinities has been employed in modern system of mathematics, it is discovered that in set theory, the thinking logic of actual infinities has not been applied consistently throughout, and that in the theory of limits, the idea of potential infinities has not been utilized consistently throughout, either. As for those subsystems involving the concepts of infinities of modern mathematics, they generally contain both kinds of infinities at the same time. As a matter of fact in modern mathematics and its theoretical foundation, one only needs to slightly analyze and dig deeper, one will see the reality that the thinking logics and method of analysis of employing both kinds of infinities are everywhere implicitly.
Originality/value
The authors show the first time in history that the system of modern mathematics is not consistent as what has been believed.
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Yi Lin, Wujia Zhu, Ningsheng Gong and Guoping Du
The paper aims to show the existence of the systemic yoyo structure in human thoughts so that the human way of thinking is proven to have the same structure as that of the…
Abstract
Purpose
The paper aims to show the existence of the systemic yoyo structure in human thoughts so that the human way of thinking is proven to have the same structure as that of the material world.
Design/methodology/approach
Parallel comparison is used to reveal the underlying structure existing in human thoughts.
Findings
After highlighting all the relevant ideas and concepts, which are behind each and every crisis in the foundations of mathematics, it becomes clear that some difficulties in the authors' understanding of nature are originated from confusing actual infinities with potential infinities, and vice versa. By pointing out the similarities and differences between these two kinds of infinities, then some hidden contradictions existing in the system of modern mathematics are handily picked out. Then, theoretically, using the authors' yoyo model, it is predicted that the fourth crisis in the foundations of mathematics has appeared. And, a plan of resolution of this new crisis is provided.
Originality/value
This paper shows the first time in history that human thought, the material world, and each economic entity, share a common structure – the systemic yoyo structure. And it proves the arrival of the fourth crisis in mathematics by using systems modeling and listing several; contradictions hidden deeply in the foundations of mathematics.
Details
Keywords
The purpose of this paper is to introduce the necessary background information and literature in order to make this special issue self‐contained.
Abstract
Purpose
The purpose of this paper is to introduce the necessary background information and literature in order to make this special issue self‐contained.
Design/methodology/approach
The paper comprises: an Introduction; Section 2 outlines the origin of the yoyo structure by presenting a brief historical account and the concept of whole evolution of systems; Section 3 introduces the literature on applications of the systemic yoyo model in areas of natural science, social science, epistemology, and practical disastrous weather forecasts; and Section 4 outlines what is contained in this special issue.
Findings
A personal account of aspects of a career in systems research and description of a personal ambition to introduce laws for social science and laws that make both natural and social sciences exact at the same time.
Originality/value
With regard to the systemic yoyo model, this paper presents an introduction to what has been accomplished by using this model and what will be presented in this special issue in order to provide all interested colleagues with an overall picture in terms of where the paper stands in the relevant research activities.
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