The Reals as Rational Cauchy Filters - NZJM

New Zealand Journal of Mathematics

Ittay Weiss¹ 

[1] School of Computing,Information and Mathematical Sciences,The University of the South Pacific,Suva, Fiji.$$

关键词: real numbers;    construction of the reals;    filter;    Cauchy filter;    minimal Cauchy filter;    rational filter;    historical survey of the real numbers;    criticism of Dedekind cuts;    criticism of Cauchy's construction.;   

DOI  :  

学科分类：社会科学、人文和艺术（综合）

来源: University of Auckland * Department of Mathematics

【 摘 要 】

We present, alongside a historical note on the development of the study of the real numbers, a detailed and elementary construction of the real numbers from the rational numbers a la Bourbaki. The real numbers are defined to be the set of all minimal Cauchy filters in (where the Cauchy condition is defined in terms of the absolute value function on ) and are proven directly, without employing any of the techniques of uniform spaces, to form a complete ordered field. The construction can be seen as a variant of Bachmann's construction by means of nested rational intervals, allowing for a canonical choice of representatives.

【 授权许可】

Unknown