A comprehensive review of the characterization of real numbers
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...
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| Format: | Article |
| Language: | Spanish |
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Universidad Nacional de Trujillo
2024-12-01
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| Series: | Selecciones Matemáticas |
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| Online Access: | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6160 |
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| author | Víctor Arturo Martínez León Rodrigo Bloot Ana Letícia de Oliveira |
| author_facet | Víctor Arturo Martínez León Rodrigo Bloot Ana Letícia de Oliveira |
| author_sort | Víctor Arturo Martínez León |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-9e9d8703a04a4dd7a7db0b0e30381fd6 |
| institution | Kabale University |
| issn | 2411-1783 |
| language | Spanish |
| publishDate | 2024-12-01 |
| publisher | Universidad Nacional de Trujillo |
| record_format | Article |
| series | Selecciones Matemáticas |
| spelling | doaj-art-9e9d8703a04a4dd7a7db0b0e30381fd62025-01-03T03:12:19ZspaUniversidad Nacional de TrujilloSelecciones Matemáticas2411-17832024-12-01110230332510.17268/sel.mat.2024.02.08A comprehensive review of the characterization of real numbersVíctor Arturo Martínez León0https://orcid.org/0000-0002-2082-6665Rodrigo Bloot1https://orcid.org/0000-0001-6504-5718Ana Letícia de Oliveira2https://orcid.org/0009-0004-4981-9575Universidade Federal da Integracao Latino-Americana (UNILA), Instituto Latino-Americano de Ciencias da Vida e da Natureza (ILACVN), Foz do Iguacu-Paraná, BrasilUniversidade Federal da Integracao Latino-Americana (UNILA), Instituto Latino-Americano de Ciencias da Vida e da Natureza (ILACVN), Foz do Iguacu-Paraná, BrasilSecretaria de Educacao do Estado do Paraná, Foz do Iguacu-Paraná, BrasilThe 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.https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6160supremum axiomcauchy sequencescomplete ordered fieldarchimedean field |
| spellingShingle | Víctor Arturo Martínez León Rodrigo Bloot Ana Letícia de Oliveira A comprehensive review of the characterization of real numbers Selecciones Matemáticas supremum axiom cauchy sequences complete ordered field archimedean field |
| title | A comprehensive review of the characterization of real numbers |
| title_full | A comprehensive review of the characterization of real numbers |
| title_fullStr | A comprehensive review of the characterization of real numbers |
| title_full_unstemmed | A comprehensive review of the characterization of real numbers |
| title_short | A comprehensive review of the characterization of real numbers |
| title_sort | comprehensive review of the characterization of real numbers |
| topic | supremum axiom cauchy sequences complete ordered field archimedean field |
| url | https://revistas.unitru.edu.pe/index.php/SSMM/article/view/6160 |
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