• Korean Journal of Logic
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• v.2
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• pp.7-33
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• 1998

최근 에체멘디는 초과생성과 미달생성의 문제를 들어 타르스키의 모델이론적 논리적 귀결의 정의의 외연적 적합성을 공격하였다. 그러한 공격의 기저에는 우연성 문제가 도사리고 있다고 보이고, 실질적으로 타르스키류의 정의를 적용함에 있이 무만 공리를 통해 논리외적 요소기 개입할 위험이 있다는 것이 그의 근본적 가정이라 생각된다. 이 글에서는 무한 공리가 논리적 진리일 기능성을 조심스레 타진이고, 논리상항과 비논리상항을 기리는 문제가 에체멘디가 생각하듯 신화가 아니라 논리적인 것과 논리외적인 것을 구별하는 문제와 동일한, 진정한 철학적 문제임을 보이는 데 노력한다.

• Journal of Educational Research in Mathematics
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• v.24 no.4
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• pp.595-605
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• 2014

Infinite decimals have an infinite number of digits, chosen arbitrary and independently, to the right side of the decimal point. Since infinite decimals are ambiguous numbers impossible to write them down completely, the infinite decimal representation accompanies unavoidable opaqueness. This article focused the transparent aspect of infinite decimal representation with respect to the completeness axiom of real numbers. Long before the formalization of real number concept in $19^{th}$ century, many mathematicians were able to deal with real numbers relying on this transparency of infinite decimal representations. This analysis will contribute to overcome the double discontinuity caused by the different conceptualizations of real numbers in school and academic mathematics.

• Journal for History of Mathematics
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• v.22 no.4
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• pp.115-128
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• 2009

In this paper, we modify the axiom of infinity of von Neumann's axioms modified by Pinter([5]), and then discuss an existence of an unknowable class in the system and a relationship between the unknowable class and the axiom of choice.

• Journal for History of Mathematics
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• v.9 no.2
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• pp.30-42
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• 1996

최근 <수학기초론>이란 용어는 Burali-Forti paradox 이후 족(class)과 집합(set) 개념을 이해하려는 시도에서 출발한 20세기적 문제에 적용되고 있다. 이 글에서는 그 해결책으로 제시된 주의ㆍ주장 중 논리적인 모순을 해결하기 위한 Russel의 논리주의적 공리론에 바탕을 두고 살펴보려고 한다. 제 2장에서는 무한의 심연 속에 웅크리고 있는 집합론에서의 역설과 발생 원인에 대하여 살펴보았다. 제 3장에서는 공리론적 집합론 중에서 러셀의 유형론과 그것을 단순화시킨 현대의 유형론을 살펴보고, ZF 집합론과 ZF 집합론의 연장인 처치 집합론의 기본 공리를 살펴보았다.

• Korean Journal of Logic
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• v.8 no.1
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• pp.23-46
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• 2005

The purpose of the paper Is to discuss and estimate early Popper's theory of probability, which is presented in his book, The Logic of of Scientific Discovery. For this, Von Mises' frequency theory shall be discussed in detail, which is regarded as the most systematic and sophisticated frequency theory among others. Von Mises developed his theory to response to various critical questions such as how finite and empirical collectives can be represented in terms of infinite and mathematical collectives, and how the axiom of randomness can be mathematically formulated. But his theory still has another difficulty, which is concerned with the inconsistency between the axiom of convergence and the axiom of randomness. Defending the objective theory of probability, Popper tries to present his own frequency theory, solving the difficulty. He suggests that the axiom of convergence be given up and that the axiom of randomness be modified to solve Von Mises' problem. That is, Popper introduces the notion of ordinal selection and neighborhood selection to modify the axiom of randomness. He then shows that Bernoulli's theorem is derived from the modified axiom. Consequently, it can be said that Popper solves the problem of inconsistency which is regarded as crucial to Von Mises' theory. However, Popper's suggestion has not drawn much attention. I think it is because his theory seems anti-intuitive in the sense that it gives up the axiom of convergence which is the basis of the frequency theory So for more persuasive frequency theory, it is necessary to formulate the axiom of randomness to be consistent with the axiom of convergence.

• Journal of Educational Research in Mathematics
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• v.26 no.2
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• pp.159-172
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• 2016

This study examines the approach of introduction of the real numbers through the infinite decimal, which is suggested by Lee Ji-Hyun(2014; 2015) in the aspect of the overcoming the double discontinuity, and analyses Li(2011), which is the mathematical background of the foregoing Lee's. Also, this study compares these construction methods given by Lee and Li with the traditional method using the nested intervals. As a result of analysis, this study shows that Lee Ji-Hyun(2014; 2015) and Li(2011) face the risk of the circulation logic in making the infinite decimal corresponding each point on the geometrical line, and need the steps not using the 'limit to a real number' in order to compensate the mathematical and educational defect. Accordingly, this study raises the opinion that the traditional method of defining the infinite decimal as a sequence by using the geometrical nested intervals axiom would be a appropriate supplementation.

• Journal of Educational Research in Mathematics
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• v.25 no.3
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• pp.263-279
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• 2015

The introduction of real numbers is one of the most difficult steps in the teaching of school mathematics since the mathematical justification of the extension from rational to real numbers requires the completeness property. The author elucidated what questions about real numbers can be unanswered as the "institutional didactic void" in school mathematics defining real numbers as the union of the rational and irrational numbers. The pre-service teachers' explanations on the extension from rational to real numbers and the raison d'$\hat{e}$tre of arbitrary non-recurring decimals showed the superficial and fragmentary understanding of real numbers. Connecting school mathematics to university mathematics via the didactic void, the author discussed how pre-service teachers could break the illusion of transparency about the real number.

• Communications of Mathematical Education
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• v.22 no.4
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• pp.471-487
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• 2008

Inventive mathematical thinking, original mathematical problem solving ability, mathematical invention and so on are core concepts, which must be emphasized in all branches of mathematical education. In particular, Polya(1981) insisted that inventive thinking must be emphasized in a suitable level of university mathematical courses. In this paper, the author considered two cases of inventive problem solving ability shown by his many students via real analysis courses. The first case is about the proof of the problem "what is the derived set of the integers Z?" Nearly all books on mathematical analysis sent the question without the proof but some books said that the answer is "empty". Only one book written by Noh, Y. S.(2006) showed the proof by using the definition of accumulation points. But the proof process has some mistakes. But our student Kang, D. S. showed the perfect proof by using The Completeness Axiom, which is very useful in mathematical analysis. The second case is to show the infinite countability of NxN, which is shown by informal proof in many mathematical analysis books with formal proofs. Some students who argued the informal proof as an unreasonable proof were asked to join with us in finding the one-to-one correspondences between NxN and N. Many students worked hard and find two singled-valued mappings and one set-valued mapping covering eight diagrams in the paper. The problems are not easy and the proofs are a little complicated. All the proofs shown in this paper are original and right, so the proofs are deserving of inventive mathematical thoughts, original mathematical problem solving abilities and mathematical inventions. From the inventive proofs of his students, the author confirmed that any students can develope their mathematical abilities by their professors' encouragements.

• The Journal of Korean Philosophical History
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• no.27
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• pp.197-222
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• 2009

The Reform Movement(戊戌變法) of 1898 was a boundary tablet of modern history of science and technology which inherited the past and ushered in the future. Kang Youwei(康有爲), as a leader, his scientific thoughts opened up the way of Chinese enlightenity campaign and pushed the development of Chinese modem science and had an important position in modem history of scientific thoughts. The dissertation analyses the source, establishment and content of Kang Youwei. Kang Youwei developed the useful and discarded the useless of the view of implement science held by the Westernized Party, undertook a deep and throughout thinking on the nature of science, had cognition of scientific methods and spirit, by which he criticized negative proneness of ancient Chinese views of science. He put forwards a series of practical suggestions on political reform that provided a solid guarantee and support in system for scientific development. Kang Youwei rooted in the soil of Chinese traditional academic culture, but also western learning in modern western civilization. Kang go through Westernization Movement since the in-depth study of Western natural and inevitable outcome of the social sciences, are giving to science and technology. Although he was originally of Western "science" has a lot of misunderstandings and prejudices, but these shallow hazy perceptual knowledge, his view of science which constitutes the basis of the formation. In the course of scientific inquiry, Kang has begun to explore the essence of scientific development. He has a gut feeling that behind the scientific discovery of the existence of a force, which is the scientific truth and is used to grasp the scientific method. After contact with the Western world, with the traditional "Heaven(天)", and modern Chinese intellectuals began to "axiom(公理)" to recover his traditional "Heaven" of the new understanding is reflected mainly in "Zhutianjiang(諸天講)". "Zhutianjiang" is the Kang Yuwei in the absorption of traditional astronomy knowledge base, will the traditional arithmetic, as well as Buddhism and the West since the twentieth century, new knowledge of astronomy combines written. Kang while recognizing that scientific instruments, is nothing more than an extension of the role of the human senses and make the "Dao(道)" is more clear, but the "artifacts(器物)" caused by the inherent limitations of the limited nature of human knowledge, which is "Heaven" boundless nature of the broad terms, refused to concede defeat to. In reality, the activities of political reform, he gradually recognize this real-world helpless, and he recognized that the real world to achieve common ground of social ideal is impossible, so he chose comfort in life that people really get a stage of "Tianyou(天遊)". This is the cause that his writing "Datongshu(大同書)", at the same time, followed by writing "Zhutianjiang" talk "Tianyou".