{"title":"理查德·戴德金的传奇命运","authors":"G. Devillanova, Giovanni Molica Bisci","doi":"10.1478/AAPP.99S1A18","DOIUrl":null,"url":null,"abstract":"By using the preliminary results given in a previous divulgative note, we present here a concise and self–contained introduction to the construction of the real field as the unique, up to increasing isomorphism, Dedekind complete totally ordered field. Moreover, we also show the equivalence between the Dedekind completeness property on totally ordered fields and some meaningful well–known notions present in the literature, such as the Cauchy completeness on totally ordered Archimedean fields. This characterization result allows us to correctly encode the Dedekind completeness for totally ordered fields in the general abstract setting of metric spaces. We believe that the essential parts of the paper can be easily accessed by anyone with some experience in abstract mathematical thinking. The paper completes the lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018.","PeriodicalId":43431,"journal":{"name":"Atti Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche Matematiche e Naturali","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2021-09-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"The fabulous destiny of Richard Dedekind\",\"authors\":\"G. Devillanova, Giovanni Molica Bisci\",\"doi\":\"10.1478/AAPP.99S1A18\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"By using the preliminary results given in a previous divulgative note, we present here a concise and self–contained introduction to the construction of the real field as the unique, up to increasing isomorphism, Dedekind complete totally ordered field. Moreover, we also show the equivalence between the Dedekind completeness property on totally ordered fields and some meaningful well–known notions present in the literature, such as the Cauchy completeness on totally ordered Archimedean fields. This characterization result allows us to correctly encode the Dedekind completeness for totally ordered fields in the general abstract setting of metric spaces. We believe that the essential parts of the paper can be easily accessed by anyone with some experience in abstract mathematical thinking. The paper completes the lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018.\",\"PeriodicalId\":43431,\"journal\":{\"name\":\"Atti Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche Matematiche e Naturali\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":0.6000,\"publicationDate\":\"2021-09-30\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Atti Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche Matematiche e Naturali\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1478/AAPP.99S1A18\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MULTIDISCIPLINARY SCIENCES\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Atti Accademia Peloritana dei Pericolanti-Classe di Scienze Fisiche Matematiche e Naturali","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1478/AAPP.99S1A18","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MULTIDISCIPLINARY SCIENCES","Score":null,"Total":0}
By using the preliminary results given in a previous divulgative note, we present here a concise and self–contained introduction to the construction of the real field as the unique, up to increasing isomorphism, Dedekind complete totally ordered field. Moreover, we also show the equivalence between the Dedekind completeness property on totally ordered fields and some meaningful well–known notions present in the literature, such as the Cauchy completeness on totally ordered Archimedean fields. This characterization result allows us to correctly encode the Dedekind completeness for totally ordered fields in the general abstract setting of metric spaces. We believe that the essential parts of the paper can be easily accessed by anyone with some experience in abstract mathematical thinking. The paper completes the lecture given by the second author during the International Workshop on New Horizons in Teaching Science in Messina on June 2018.
期刊介绍：
This journal is of a multi- and inter-disciplinary nature and covers a broad range of fields including mathematics, computer science, physics, chemistry, biology, earth sciences, and their intersection. History of science is also included within the topics addressed by the journal. The transactions of the Pelorian Academy started out as periodic news sheets containing the notes presented by the members of the Divisions into which the Academy has been and still is organized, according to subject areas. The publication of these notes for the Division (“Classe”) of Mathematical, Physical and Natural Sciences is the responsibility of the Editorial Committee, which is composed of the Director of the division with the role of Chairman, the Vice-Director, the Secretary and two or more other members. Besides original research articles, the journal also accepts texts from conferences and invited talks held in the Academy. These contributions are published in a different section of the journal. In addition to the regular issues, single monographic supplements are occasionally published which assemble reports and communications presented at congresses, symposia, seminars, study meetings and other scientific events organized by the Academy or under its patronage. Since 2004 these transactions have been published online in the form of an open access electronic journal.
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