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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. [ABSTRACT FROM AUTHOR]

Purpose - The paper aims to show using a different method that any uncountable set is a self-contradictory non-set. Design/methodology/approach - The paper discusses the concept. Findings - Elsewhere it is shown that in the framework of ZFC, various countable infinite sets are all self-contradicting non-sets. In this paper, the authors will generalize the concept of Cauchy theater, and establish the concept of transfinite Cauchy theaters. After that, employing a new method, they will prove that various uncountable infinite sets, as studied in naive set theory and modern axiomatic set theory, are also self-contradicting non-sets. Originality/value - The concept of general Cauchy theater is introduced. [ABSTRACT FROM AUTHOR]

Purpose - The paper aims to employ a different approach to show that the countable infinite sets are self-contradictory non-sets. Design/methodology/approach - The paper discusses the concept. Findings - The concept of infinities in the countable set theory was discussed in Zhu et al. by employing the method of analysis of allowing two different kinds of infinities. What was obtained is that various countable infinite sets, studied in the naïve and modern axiomatic set theories, are all incorrect concepts containing self-contradictions. In this paper, the authors provide another argument to prove the same conclusion: various countable infinite sets studied in both naïve and modern axiomatic set theories are all specious non-sets. The argument is given from a different angle on still the same premise of allowing two different concepts of infinities. Originality/value - The concept of Cauchy theater is introduced. [ABSTRACT FROM AUTHOR]

Purpose - The purpose is to show that all uncountable infinite sets are self-contradictory non-sets. Design/methodology/approach - A conceptual approach is taken in the paper. Findings - Given the fact that the set N={x|n(x)} of all natural numbers, where n(x)=df "x is a natural number" is a self-contradicting non-set in this paper, the authors prove that in the framework of modern axiomatic set theory ZFC, various uncountable infinite sets are either non-existent or self-contradicting non-sets. Therefore, it can be astonishingly concluded that in both the naive set theory or the modern axiomatic set theory, if any of the actual infinite sets exists, it must be a self-contradicting non-set. Originality/value - The first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]

Purpose - On the basis of previous work, the authors aim to further study the problem of infinity existing in between predicates and sets. Design/methodology/approach - A conceptual approach is taken in the paper. Findings - The authors modify the conventional rule of thinking that each predicate determines a unique set, and establish a principle regarding the relationship between predicates and sets. Then, the authors study the structures of actually infinite, rigid sets. Originality/value - The structure of actually infinite sets is detailed. [ABSTRACT FROM AUTHOR]

Purpose - The paper's aim is to reconsider the feasibility at both the heights of mathematics and philosophy of the statement that each predicate determines a unique set. Design/methodology/approach - A conceptual approach is taken. Findings - In the naive and the modern axiomatic set theories, it is a well-known fact that each predicate determines precisely one set. That is to say, for any precisely defined predicate P, there is always A={x|P(x)} or x?A?P(x). However, when the authors are influenced by the thinking logic of allowing both kinds of infinities and compare these two kinds of infinities, and potentially infinite and actually infinite intervals and number sets, it is found that the expressions of these number sets are not completely reasonable. Originality/value - Detailed analyses are given for the introduction of new symbols and representations for either potential or actual infinite sets. [ABSTRACT FROM AUTHOR]

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. [ABSTRACT FROM AUTHOR]