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The contents of the Least upper bound axiom page were merged into Completeness of the real numbers. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
Who or what is dedekind and what is his terminology? — Preceding unsigned comment added by 2601:58B:4204:B6B0:FDAB:4871:5C6C:43D9 ( talk) 00:19, 7 February 2024 (UTC)
The page claims that Dedekind's formulation is used in a "synthetic approach" to the real numbers. There is no reference. What is meant by "synthetic" here? Is this related to projective geometry? To synthetic differential geometry? To something else? Tkuvho ( talk) 11:02, 10 January 2011 (UTC)
When I learned the definition of the real numbers using Cauchy sequences, we started by defining a relation on Cauchy sequences of rational numbers that was equivalent to "converges to the same real number," but did not use the notions of "real number" or "converges." It was something like "the tails cannot be bounded away from one another," but I don't have the book in front of me right now. Then we proved that it was an equivalence relation, defined real numbers as the set of equivalence classes, defined operations on real numbers in terms of those equivalence classes, and proved that the real numbers, so defined, had all the expected properties. I'm not sure about the bit "Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers," in the section "Cauchy completeness," since it seems to obscure this. Is there some other way to define the reals with Cauchy sequences that works differently, or is this part of the page deliberately omitting detail about equivalence classes, perhaps due to assumptions about the audience of the page? — Preceding unsigned comment added by 50.58.96.2 ( talk) 17:43, 7 November 2017 (UTC)
The Heine-Borel Theorem is another form of completeness that is often used in analysis that probably should be added to this article. I would add it myself but I am unsure whether it is itself equivalent to the other conditions or whether it needs the Archimedean property. — Preceding unsigned comment added by Joshuatmeadows ( talk • contribs) 21:02, 22 January 2018 (UTC)
"The intermediate value theorem states that every continuous function that attains both negative and positive values has a root."
It is Bolzano's theorem, not the intermediate value theorem that states this. Kontribuanto ( talk) 16:58, 26 February 2024 (UTC)
This
level-5 vital article is rated C-class on Wikipedia's
content assessment scale. It is of interest to the following WikiProjects: | |||||||||||
|
The contents of the Least upper bound axiom page were merged into Completeness of the real numbers. For the contribution history and old versions of the redirected page, please see its history; for the discussion at that location, see its talk page. |
Who or what is dedekind and what is his terminology? — Preceding unsigned comment added by 2601:58B:4204:B6B0:FDAB:4871:5C6C:43D9 ( talk) 00:19, 7 February 2024 (UTC)
The page claims that Dedekind's formulation is used in a "synthetic approach" to the real numbers. There is no reference. What is meant by "synthetic" here? Is this related to projective geometry? To synthetic differential geometry? To something else? Tkuvho ( talk) 11:02, 10 January 2011 (UTC)
When I learned the definition of the real numbers using Cauchy sequences, we started by defining a relation on Cauchy sequences of rational numbers that was equivalent to "converges to the same real number," but did not use the notions of "real number" or "converges." It was something like "the tails cannot be bounded away from one another," but I don't have the book in front of me right now. Then we proved that it was an equivalence relation, defined real numbers as the set of equivalence classes, defined operations on real numbers in terms of those equivalence classes, and proved that the real numbers, so defined, had all the expected properties. I'm not sure about the bit "Essentially, this method defines a real number to be the limit of a Cauchy sequence of rational numbers," in the section "Cauchy completeness," since it seems to obscure this. Is there some other way to define the reals with Cauchy sequences that works differently, or is this part of the page deliberately omitting detail about equivalence classes, perhaps due to assumptions about the audience of the page? — Preceding unsigned comment added by 50.58.96.2 ( talk) 17:43, 7 November 2017 (UTC)
The Heine-Borel Theorem is another form of completeness that is often used in analysis that probably should be added to this article. I would add it myself but I am unsure whether it is itself equivalent to the other conditions or whether it needs the Archimedean property. — Preceding unsigned comment added by Joshuatmeadows ( talk • contribs) 21:02, 22 January 2018 (UTC)
"The intermediate value theorem states that every continuous function that attains both negative and positive values has a root."
It is Bolzano's theorem, not the intermediate value theorem that states this. Kontribuanto ( talk) 16:58, 26 February 2024 (UTC)