Correct a Wide Spread Conclusion of Cantor Set Theory
Author(s)
Ming Xiong
ABSTRACT
The Cantor’s conclusion that “there is a one-one correspondence between the points on n-dimensional space and the points on the line” appears in the numerous current documents. By introducing a monotonic and continuous function, a one-one correspondence between two intervals is built; and by using parametric equations of the curve, a one-one correspondence from the points on the curve to the points on the line is established. Specially, the meanings of multivariate functions are given. By using a n-variable equation with a parameter, a correspondence from n-dimensional space area to a interval is built, so the wrong conclusion is completely denied. The paper enriches calculus and can reduce the teaching difficulty of real function in some degree. The expression of moving curve (surface) limit is given in the paper. More importantly, after the conclusion is corrected, it will be necessary and possible to re-establish the theory and the approach about multivariable differential calculus.
The Cantor’s conclusion that “there is a one-one correspondence between the points on n-dimensional space and the points on the line” appears in the numerous current documents. By introducing a monotonic and continuous function, a one-one correspondence between two intervals is built; and by using parametric equations of the curve, a one-one correspondence from the points on the curve to the points on the line is established. Specially, the meanings of multivariate functions are given. By using a n-variable equation with a parameter, a correspondence from n-dimensional space area to a interval is built, so the wrong conclusion is completely denied. The paper enriches calculus and can reduce the teaching difficulty of real function in some degree. The expression of moving curve (surface) limit is given in the paper. More importantly, after the conclusion is corrected, it will be necessary and possible to re-establish the theory and the approach about multivariable differential calculus.
Cite this paper
Xiong, M. (2015) Correct a Wide Spread Conclusion of Cantor Set Theory. Advances in Pure Mathematics, 5, 544-551. doi: 10.4236/apm.2015.59050.
Xiong, M. (2015) Correct a Wide Spread Conclusion of Cantor Set Theory. Advances in Pure Mathematics, 5, 544-551. doi: 10.4236/apm.2015.59050.
References
[1] Ling, L.J. (2007) Selected Mathematical History (Optional Courses 3-1). The People’s Education Press, Beijin, 71-76. [2] Zhang, H. (2012) Brief History of Mathematics. Science Press, Beijin, 227. [3] Cheng, C.D. (2012) Introduction of Real Function. Science Press, Beijin, 17. [4] Zhou, X.W. and Sun, W.C. (2014) Real Function. Science Press, Beijin, 12. [5] Tang, Y.Z. (2007) A Method of Changing Multiple Intefrals and Surface Integrals Directly into Simple Integrals. Studies in college Mathematics, 10, 10. [6] Li, G.Q. and Li, S.Z. (2009) Mathematics (Volume One). Higher Education Press, Beijin, 19. [7] Xiong, M. and Xiong, J. (2012) Infiltrating the thought of Limit in Teaching Real Number. Course Education Research, 1, 64. [8] Xiong, M. (2009) Construction of the Regional Elements of Differential Calculus and Re-integration into a Single-integral Directly. Journal of Sichuan College of Education, 25, 32. [9] Xiong, M. (2010) Change Double Integral into Simple Integral with Area Elment. Studies in College Mathematics, 13, 115. [10] Xiong, M. (2010) Eight Propositions for Simplifing Calcuation of Multiintegrals. Journal of Southwest University for Nationalities (Natural Science Edition), 36, 595. [11] Xiong, M. (2013) Moving Curve (Surface) and Degenerate Transformation for Multi-Integrals. Studies in College Mathematics, 16, 22.
[1] Ling, L.J. (2007) Selected Mathematical History (Optional Courses 3-1). The People’s Education Press, Beijin, 71-76. [2] Zhang, H. (2012) Brief History of Mathematics. Science Press, Beijin, 227. [3] Cheng, C.D. (2012) Introduction of Real Function. Science Press, Beijin, 17. [4] Zhou, X.W. and Sun, W.C. (2014) Real Function. Science Press, Beijin, 12. [5] Tang, Y.Z. (2007) A Method of Changing Multiple Intefrals and Surface Integrals Directly into Simple Integrals. Studies in college Mathematics, 10, 10. [6] Li, G.Q. and Li, S.Z. (2009) Mathematics (Volume One). Higher Education Press, Beijin, 19. [7] Xiong, M. and Xiong, J. (2012) Infiltrating the thought of Limit in Teaching Real Number. Course Education Research, 1, 64. [8] Xiong, M. (2009) Construction of the Regional Elements of Differential Calculus and Re-integration into a Single-integral Directly. Journal of Sichuan College of Education, 25, 32. [9] Xiong, M. (2010) Change Double Integral into Simple Integral with Area Elment. Studies in College Mathematics, 13, 115. [10] Xiong, M. (2010) Eight Propositions for Simplifing Calcuation of Multiintegrals. Journal of Southwest University for Nationalities (Natural Science Edition), 36, 595. [11] Xiong, M. (2013) Moving Curve (Surface) and Degenerate Transformation for Multi-Integrals. Studies in College Mathematics, 16, 22.