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PurposeThe 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/approachA conceptual approach is taken.FindingsIn 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/valueDetailed analyses are given for the introduction of new symbols and representations for either potential or actual infinite sets.

PurposeThe paper aims to show using a different method that any uncountable set is a self‐contradictory non‐set.Design/methodology/approachThe paper discusses the concept.FindingsElsewhere 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/valueThe concept of general Cauchy theater is introduced.

PurposeThe paper aims to introduce the concepts of potential and actual infinities.Design/methodology/approachA conceptual approach is taken.FindingsIt 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/valueThe authors show the first time in history that the system of modern mathematics is not consistent as what has been believed.

PurposeOn the basis of previous work, the authors aim to further study the problem of infinity existing in between predicates and sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsThe 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/valueThe structure of actually infinite sets is detailed.

PurposeThe paper's aim is to show a pair of deeply hidden contradictions in the system of mathematics.Design/methodology/approachThe paper takes a conceptual approach to the problem.FindingsIt 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/valueBy using the concepts of potential and actual infinities, the authors uncover two contradictory thinking logics widely employed in the study of mathematics.

PurposeThe purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.Design/methodology/approachA conceptual approach is taken in the paper.FindingsGiven 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/valueThe first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

PurposeThe paper's aim is to reveal the return of the Berkeley paradox of the eighteenth century.Design/methodology/approachThis is a conceptual discussion.FindingsSince, long time ago, the common belief has been that the establishment and development of the theory of limits had provided an explanation for the Berkeley paradox. However, when the authors revisit some of the age‐old problems using the thinking logic of allowing both the concepts of potential and actual infinities, they find surprisingly that the shadow of the Berkeley paradox does not truly disappear in the foundation of mathematical analysis.Originality/valueShow the incompleteness of the theory of limits, which is not the same as what has been believed in the history of mathematics.

PurposeThe paper aims to employ a different approach to show that the countable infinite sets are self‐contradictory non‐sets.Design/methodology/approachThe paper discusses the concept.FindingsThe 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/valueThe concept of Cauchy theater is introduced.

PurposeAims to focus on the co‐existence of potential and actual infinities in modern mathematics and its theoretical foundation. It has been shown that not only the whole system of modern mathematics but also the subsystems directly dealing with infinities have permitted the co‐existence of these two kinds of infinities.Design/methodology/approachThe paper discusses the issues surrounding the two problems that urgently need to be solved. One of the problems is how to select an appropriate theoretical foundation for modern mathematics and the theory of computer science. The other problem is, under what interpretation can modern mathematics and the theory of computer science be kept in their entirety?FindingsThis paper constructs the mathematical system of potential infinities in an effort to address the two afore‐mentioned problems.Originality/valueHighlights that the said mathematical system of potential infinities is completely different of the mathematical system constructed on the basis of intuitionism.

PurposeThe paper aims to use a third method to show that the system of natural numbers is inconsistent.Design/methodology/approachA conceptual approach is taken.FindingsWithout directly employing the concepts of potential and actual infinities, the authors show that the concept of the set N={x|n(x)}, where n(x)=def “x is a natural number” of all natural numbers is a self‐contradicting, incorrect concept.Originality/valueThe paper shows the system of natural numbers to be inconsistent.

PurposeThe 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/approachParallel comparison is used to reveal the underlying structure existing in human thoughts.FindingsAfter 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/valueThis 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.

PurposeThe paper aims to show countable infinite sets are self‐contradictory non‐sets.Design/methodology/approachThe paper is a conceptual discussion.FindingsSince, 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/valueA well‐known concept is shown to be not as correct as what has been believed.

PurposeThis paper is the third part of the effort to resolve the following two problems, which 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/approachThe paper is a conceptual discussion of the metatheory.FindingsThe paper establishes the metatheory of the logical foundation for the mathematical system of potential infinities.Originality/valueThe authors prove the relevant results on the reliability and the completeness of the logical system.

Purpose – This is the second part of the effort to resolve the following two problems that badly 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 sets out the foundation for the system.Findings – Here, the logical foundation for the mathematical system of potential infinities is given.Originality/value – The logical calculus, which will be used as the tool of deduction in the PIMS, is established. This new tool of reasoning is a modification of the classical two‐value logical calculus system.

PurposeThe paper's purpose is to analyze the concepts of potential and actual infinities.Design/methodology/approachThe 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.FindingsClarify 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/valueIt 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.

PurposeThe paper's purpose is to show a class of well‐used methods in set theory is invalid.Design/methodology/approachA conceptual approach is taken.FindingsBased 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/valuePoint out more paradoxes in the methodology of mathematics.

PurposeThis 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/approachThe paper is a conceptual discussion.FindingsThe paper lays out the set theoretical foundation for the mathematical system of potential infinities.Originality/valueThis 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.