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On the basis of previous work, the authors aim to further study the problem of infinity existing in between predicates and sets.

A conceptual approach is taken in the paper.

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.

The structure of actually infinite sets is detailed.

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 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.

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.

The paper aims to show using a different method that any uncountable set is a self‐contradictory non‐set.

The paper discusses the concept.

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.

The concept of general Cauchy theater is introduced.

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'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'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 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.

Aims 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.

The 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?

This paper constructs the mathematical system of potential infinities in an effort to address the two afore‐mentioned problems.

Highlights that the said mathematical system of potential infinities is completely different of the mathematical system constructed on the basis of intuitionism.