Chinese Quarterly Journal of Mathematics ›› 1998, Vol. 13 ›› Issue (4): 24-28.

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An Equivalent Form of the Dedekind Axiomand Its Application(Ⅱ)——Also on the unity of the Continuous Induction,the Mathem atical Induction and the Transfinite Induction

  

  1.  Department of Mathematics，Guangzhou Teacherʾs College，Guangzhou，510400
  • Received:1997-11-20 Online:1998-12-30 Published:2024-10-14
  • Supported by:
    Project supported by the following：① the1352Project Fund of ninth five-year plan made by Higher Education Office of
    Guangdong Province．② the Subjec-t supporting Fund for Colleges and University of Guangdong Province．③ the Important
    Subject Fund for Colleges and Universities of Guangzhou．T he project is included in National Educational Projects of1998－
    2000．

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Abstract:  In this paper，a common characteristic of real number system and well ordered set is revealed and proved to be an quivalent form of the Dedekind Axiom or the Continuous Induction in R．Basing on it，we get the unified form of the mathematical induction，the transfinite induction and the continuous induction and generalize induction to totally ordered set that has the same characteristic．

Key words: mathematical induction, transfinite induction, continuous induction, well order, totally order, Dedekind?s Axiom

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