Talk:Archimedean property

message: Should it be "Archimedean" or "archimedean"? Note that archimedean field uses the downcase version. Oleg Alexandrov 01:25, 13 Feb 2005 (UTC)
replies: text: I'm going to post this to WP:MATH. - grubber 16:02, 18 April 2007 (UTC)
thread_title:  Capitalized or not? 

message: This article purported to be very general, speaking of arbitrary algebraic structures, but actually, it didn't encompass probably the most important case of non-Archimedean fields (which are usually not ordered!). I am not convinced that the ultimate generality is warranted, and I've begun the clean-up with a slightly more intuitive lead, and a narrower definition in the context of linearly ordered groups, where it can be made precise. I'd like to see natural examples of non-Archimedean objects not related to non-Archimedean fields. The example of the ordered field of of rational functions is nice, but why not just give an example of the ring of formal power series (make it ordered, if you like): it would bring out the same point without the extra complexity. If there are no objections, I would replace the example. Arcfrk 05:56, 13 May 2007 (UTC)
thread_title:  Clean-up 

message: It is also interesting to note that the definition of infinitesimal isn't really constructive. Given positive number, there's is no terminating algorithm that can decide if the number is infinitesimal. Algorithm can only terminate if it finds big enough natural number n. Tlepp (talk) 13:03, 1 April 2008 (UTC)
replies: text: That is fallacious.  You're only proving that ONE PARTICULAR algorithm doesn't terminate, but you're claiming to have proved there IS NO ALGORITHM that terminates, i.e. you're claiming to have proved there is no OTHER algorithm besides that one.  Obviously in the example of rational functionss over an ordered field, to claim that certain rational fuunctions are infinitesimal is completely constructive. Michael Hardy (talk) 13:13, 1 April 2008 (UTC)
replies: text:  I not proving nor claiming to have proved anything. I just made comment about the generic definition of infinitesimal in the most generic case of ordered group. I found it interesting that constructivist even care about Archimedean property of reals. Intuitively constructivism and infinitesimals seem to be far apart. Tlepp (talk) 14:19, 1 April 2008 (UTC)

text: You claimed there is no terminating algorithm.  That is wrong.  You said an algorithm can only terminate if it finds big enough natural number n.  That is wrong.  That would apply ONLY to an algorithm that proceeds just by computing nx for n = 1, 2, 3, ... or the like.  The fact that that one sort of algorithm won't terminate if x is infinitesimal does not mean that NO OTHER algorithm will terminate.  There's nothing at all non-contructive about the example involving rational functions over an ordered field. Michael Hardy (talk) 15:26, 1 April 2008 (UTC)
thread_title:  Archimedean property in constructive analysis 

message: I reverted this edit for a couple reasons:
*It (probably accidentally) removed the part of the proof that deals with the possibility of c being positive and not infinitesimal.
*There is no need to explicitly deal with the possibility of c being negative; Z includes 0, so c is at least 0.
*The language "increases the size of Z" is visually appealing, but it could mislead the reader who takes it at face value.
Melchoir (talk) 07:20, 3 August 2008 (UTC)
replies: text: So you reverted it instead of fixing it? The reason I changed the wording is that it could be more comprehensible to more of us who haven't seen this term or had it explained to us in any class we've taken. The way it's written now is more like a review of what you might have already seen but forgot. The way I rewrote it made it more accessible to someone who had never seen it before. Please rewrite that section again according to your standards, please, making it more readable to others who would like to learn this term. D. F. Schmidt (talk) 04:56, 5 August 2008 (UTC)
replies: text: Okay. I restructured the proof. The good news: it's more streamlined. The bad news: the new version is less explicit than the previous version, as it doesn't actually apply the definition of "infinitesimal", and it assumes some intuitive facts like the ordering of infinitesimals without proof. I think it's still rigorous enough for the purposes of this article. Melchoir (talk) 01:14, 6 August 2008 (UTC)
replies: text:  Thanks a lot. I think it will help, and the relationship to the Archimedean field might be a lot more helpful than that simple (shoddy, imo) explanation that was there before. Now if there are further questions, it appears they might belong elsewhere. D. F. Schmidt (talk) 06:44, 6 August 2008 (UTC)
replies: text: Well, I don't exactly share your views on these articles... but you're welcome! Melchoir (talk) 00:30, 8 August 2008 (UTC)
thread_title:  Proof revision 

message: In the real-number proof, the step "Since c is an upper bound of Z, 2c must be non-infinitesimal" needs to be made clearer for non-mathematicians coming from the 0.999... article. -- The Anome (talk) 13:05, 9 December 2008 (UTC)
replies: text: I have added some more explanation.  Is this what you had in mind? siℓℓy rabbit (talk) 13:16, 9 December 2008 (UTC)
replies: text: Yes! Thank you. -- The Anome (talk) 18:01, 9 December 2008 (UTC)
thread_title:  Clarification needed 

message: The group G is Archimedean if there is no pair x,y such that x is infinitesimal with respect to y. That is to say, for all positive elements x and y there is a positive integer n such that nx is greater than y. 
Right now it says that G is Archimedian if there is no (positive) pair x,y such that for all n nx<y. That is absolutely correct but it is like saying  the element x generates the (additive) group G  if there is no element y such that for all integers n nx≠y. Infinitesimals are great but one can understand what Archimedian means before understanding how it can fail. --Gentlemath (talk) 23:59, 15 March 2010 (UTC)
thread_title:  Any objections to this addition? 

message: In this wikipedia article about the Archimedean axiom, nowhere was the Archimedean axiom actually stated! I added it in the first paragraph, as well as a reference to the person who first coined the term "Archimedean axiom". Mollwollfumble (talk) 19:31, 27 July 2011 (UTC)
replies: text: I removed your symbolic additions. They did not accurately describe your cited, English language addition. Cliff (talk) 21:52, 27 July 2011 (UTC)
replies: text: I'm not sure your addition belongs in the lead. Most mathematics articles have a definitions section that follows the lead. There are two definitions on this page, other than yours, that define the property in different settings. Anybody else have an opinion on whether the statement added by Mollwollfumble belongs in the lead?  Cliff (talk) 18:58, 11 August 2011 (UTC)

text: See Wikipedia:Manual of Style (mathematics)#Article introduction and Wikipedia:Manual of Style (mathematics)#Article body. It seems to say that symbolic statements should be avoided in the lead and that exact definitions belong in a definitions section following the lead. If no objections, I'll remove the statement from the lead. Cliff (talk) 19:06, 11 August 2011 (UTC)
replies: text: Definition removed from lead. Cliff (talk) 12:55, 22 August 2011 (UTC)
thread_title:  Archimedean axiom added 

message: Archimedean principle redirects here. It should perhaps redirect to Archimedes' principle, a totally different thing.--Mongreilf (talk) 21:28, 14 November 2011 (UTC)
replies: text: Tricky. From things I've seen (but I am notoriously bad at being able to provide citations) "Archimedes' Principle" is that thing with water and eureka and all that, but "the Archimedean principle" is that property held by real numbers, and generalises to spaces for which that property holds, whence it's called "the Archimedean property". So yes, in theory it should but in practice I don't think it does. --Matt Westwood 22:23, 14 November 2011 (UTC)
replies: text: disambig on Archimedian rinciple then?--Mongreilf (talk) 22:39, 14 November 2011 (UTC)
replies: text: Shrug, probably, yeah, that would work for me: Archimedean Principle --> Archimedes' Principle (Physics), Archimedean Property (Mathematics)? I think you're right - there's enough confusion here that it needs to be "something"ed. --Matt Westwood 22:51, 14 November 2011 (UTC)
replies: text: No need to rename anything. Just change the redirect page to a disambig, which I've done here Archimedean principle.--Mongreilf (talk) 13:22, 15 November 2011 (UTC)
thread_title:  Archimedean Principle 

message: The Archimedean property/principle has nothing to do with "real" numbers. Archie only acknowledged rational numbers or incommensurable magnitudes.
Also, Bk V, Def. 4 is NOT the Archimedean property. It's not even related.
To understand what the property means for the first time in your life, go here:
https://www.youtube.com/watch?v=_vO_GkC54vM
197.79.1.216 (talk) 18:07, 22 October 2015 (UTC)
thread_title:  This Wikipedia entry is completely irrelevant 

message: Trying to understand what is said in the article.
IIUC, every totally ordered Archimedean field has a subfield isomorphic to the rational numbers ℚ,+,·,≤ and is itself isomorphic to some subfield of the real numbers ℝ,+,·,≤, is that it? So in some sense one could say that (modulo an isomorphism) ℚ is "the smallest" totally ordered Archimedean field and ℝ is "the biggest"? (And IIRC, the set of real algebraic numbers with the usual +,·,≤ operations is another one, somewhere in between and maybe too little talked about in elementary algebra courses) — Tonymec (talk) 16:08, 25 July 2021 (UTC)
thread_title:  Properties