A Historical Study on the Interaction of the Limit-the Infinite Set and Its Educational Implications

극한과 무한집합의 상호작용과 그 교육적 시사점에 대한 역사적 연구

  • Received : 2017.12.04
  • Accepted : 2018.03.18
  • Published : 2018.04.30


This study begins with the awareness of problem that the education of mathematics teachers has failed to link the limit and the infinite set conceptually. Thus, this study analyzes the historical and reciprocal development of the limit and the infinite set, and discusses how to improve the education of these concepts and their relation based on the outcome of this analysis. The results of the study confirm that the infinite set is the historical tool of linking the limit and the real numbers. Also, the result shows that the premise of 'the component of the straight line is a point.' had the fundamental role in the construction of the real numbers as an arithmetical continuum and that the moral certainty of this premise would be obtained through a thought experiment using an infinite set. Based on these findings, several proposals have been made regarding the teacher education of awakening someone to the fact that 'the theoretical foundation of the limit is the real numbers, and it is required to introduce an infinite set for dealing with the real numbers.' in this study. In particular, by presenting one method of constructing the real numbers as an arithmetical continuum based on a thought experiment about the component of the straight line, this study opens up the possibility of an education that could get the limit values psychologically connected to the infinite set in overcoming the epistemological obstacle related to the continuum concept.


  1. M. ANDERSON, V. KAATZ, R. WILSON, Who gave you the epsilon?, Washington, DC, Mathematical Association of America, 2009.
  2. J. L. BELL, The Continuous and the Infinitesimal in Mathematics and Philosophy, Milano, Polimetrica, 2006.
  3. U. BOTTAZZINI, The Higher calculus: A History of Real and Complex Analysis From Euler to Weierstrass, New York, Springer-Verlag, 1986.
  4. C. B. BOYER, The History of the Calculus and Its Conceptual Development, New York, Dover Publications, 1949. 김경화 역, 미적분학사-그 개념의 발달, 서울, 교우사, 2004.
  5. C. B. BOYER, A History of Mathematics, Wiley, 1991. 양영오, 조윤동 역, 수학의 역사(하), 경문사, 2000.
  6. R. CALINGER, Classics of Mathematics, New Jersey, Prentice Hall, 1995.
  7. R. DEDEKIND, Essays on the theory of numbers, New York, Dover Publications, 1963.
  8. H. EVES, Foundations and Fundamental Concepts of Mathematics, Boston, Pws-Kent, 1990. 허민, 오혜영 역, 수학의 기초와 기본개념, 경문사, 1999.
  9. W. B. EWALD, From Kant to Hilbert, A Source Book in the Foundations of Mathematics, New York, Oxford University Press, 1999.
  10. I. GRATTAAN-GUINNESS, The Development of the Foundations of Mathematical Analysis From Euler to Riemann, Cambridge, MIT Press, 1970.
  11. I. GRATTAN-GUINNESS(ed.), From the Calculus to Set Theory, London, Duckworth, 1980.
  12. R. HERSH, What is Mathematics, Really?, New York, Oxford University Press, 1997. 허민 역, 도대체 수학이란 무엇인가? 서울, 경문사, 2003.
  13. KIM, N. H. et al, Mathematics Curriculum and a Study of Teaching Materials, Seoul, KyungMoon, 2017. 김남희 외, 수학교육과정과 교재연구, 서울, 경문사, 2017.
  14. KIM, Y. W., KIM, Y. K., Set Theory and Mathematics, Seoul, WooSung, 1991. 김용운, 김용국, 집합론과 수학, 서울, 우성문화사, 1991.
  15. G. LAKOFF, R. E. NUNEZ, Where Mathematics Comes From, New York, Basic Books, 2000.
  16. C. MCLARTY, Defining sets as sets of space, Journal of Philosophical Logic 17 (1988), 75-90.
  17. H. WEYL, Philosophy of Mathematics and Natural Science, Princeton University Press, 1947. 김상문 역, 수리철학과 과학철학, 서울, 민음사, 1990.