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The purpose is to show that all uncountable infinite sets are self‐contradictory non‐sets.

A conceptual approach is taken in the paper.

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.

The first time in history, it is shown that such convenient notion as the set of all real numbers needs to be reconsidered.

The paper aims to employ a different approach to show that the countable infinite sets are self‐contradictory non‐sets.

The paper discusses the concept.

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.

The concept of Cauchy theater is introduced.

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?

The paper is a conceptual discussion.

The paper lays out the set theoretical foundation for the mathematical system of potential infinities.

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.

This 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?

The paper is a conceptual discussion of the metatheory.

The paper establishes the metatheory of the logical foundation for the mathematical system of potential infinities.

The authors prove the relevant results on the reliability and the completeness of the logical system.

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?

The paper sets out the foundation for the system.

Here, the logical foundation for the mathematical system of potential infinities is given.

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.

The paper aims to use a third method to show that the system of natural numbers is inconsistent.

A conceptual approach is taken.

Without 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.

The paper shows the system of natural numbers to be inconsistent.

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.

A conceptual approach is taken.

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.

Detailed analyses are given for the introduction of new symbols and representations for either potential or actual infinite sets.

The paper aims to show countable infinite sets are self‐contradictory non‐sets.

The paper is a conceptual discussion.

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.

A well‐known concept is shown to be not as correct as what has been believed.

The paper's aim is to reveal the return of the Berkeley paradox of the eighteenth century.

This is a conceptual discussion.

Since, 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.

Show the incompleteness of the theory of limits, which is not the same as what has been believed in the history of mathematics.

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

The paper takes a conceptual approach to the problem.

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.”

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