Talk:Completeness of the real numbers

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)

synthetic?

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)

The word "synthetic" is used throughout mathematics to refer to any approach to a subject that begins with a list of axioms, similar to synthetic geometry. Thus a "synthetic approach" to the real numbers involves defining the real numbers using a list of axioms. This is distinguished from a constructive approach, where the real numbers are defined in terms of existing objects such as sets and functions. Jim.belk (talk) 18:01, 10 January 2011 (UTC)

I would have thought that's called an axiomatic approach, as in Hilbert's axiomatic approach to geometry and to the real numbers. Synthetic geometry (first half of 19th century) is a much older subject than the axiomatic spirit (late second half of 19th century), and is characterized not so much by axioms as by using geometric constructions rather than computations. Tkuvho (talk) 19:52, 10 January 2011 (UTC)

"Synthetic" and "axiomatic" are roughly synonyms. The idea of an axiomatic approach to geometry arguably goes back to Euclid. Jim.belk (talk) 07:09, 11 January 2011 (UTC)

Dedekind's approach is not axiomatic. It is a construction in the context of classical set theory. It may not please a constructivist, but that's a different issue. The claim currently contained in the page is in error. Tkuvho (talk) 19:54, 10 January 2011 (UTC)

You are confusing the construction of the real numbers using Dedekind cuts with the notion of Dedekind completeness. The Dedekind completeness axiom is usually assumed in the axiomatic approach to the real numbers. Jim.belk (talk) 07:09, 11 January 2011 (UTC)

OK thanks. Tkuvho (talk) 14:15, 11 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)

Heine-Borel Theorem

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)

wrong theorem cited

"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)