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A comprehensive review of the characterization of real numbers

A comprehensive review of the characterization of real numbers

Authors :
Víctor Arturo Martínez León
Rodrigo Bloot
Ana Letícia de Oliveira
Source :
Selecciones Matemáticas, Vol 11, Iss 02, Pp 303-325 (2024)
Publication Year :
2024
Publisher :
Universidad Nacional de Trujillo, 2024.

Abstract

The real number system is a fundamental tool for rigorous demonstrations of the differential and integral calculus results. Even after a century of formalization on solid foundations, discussions about the construction of this field are generally omitted in advanced courses such as Real Analysis. In the present work, we present a comprehensive review on the construction and characterization of the real numbers field. The presentation focuses on the construction through Cauchy sequences of rational numbers. The notion of completeness is delimited differently from completeness when Dedekind’s cut construction is used. The results indicate Q and R Archimedean as a necessary condition for these two notions of completeness to be equivalent. To illustrate this, inspired by the work of Leon W. Cohen and Gertrude Ehrlich, we present an example of a Cauchy-complete non-Archimedean ordered field in which the supremum axiom is not equivalent to the nested intervals principle.

Details

Language :
Spanish; Castilian
ISSN :
24111783
Volume :
11
Issue :
02
Database :
Directory of Open Access Journals
Journal :
Selecciones Matemáticas
Publication Type :
Academic Journal
Accession number :
edsdoj.9e9d8703a04a4dd7a7db0b0e30381fd6
Document Type :
article
Full Text :
https://doi.org/10.17268/sel.mat.2024.02.08