Talk:Archimedean property

Early comments

Why can't 3c/4 be an infinitesimal? (I'm not saying it is...just that it's not obvious that every number less than an infinitesimal is infinitesimal, at least not obvious to those who don't work with infinitesimals...)

Okay, after a moment's thought, it IS obvious, but it still seems this fact should be mentioned (it's not even mentioned, although it's a key point in the proof).

---

How can anything be an infintesimal. Even the smallest number when summed will eventually exceed 1. Hmm...

As explained in the article, "The non-existence of nonzero infinitesimal real numbers follows from the least upper bound property of the real numbers". So you are absolutely right: There are no real infinitesimals (except 0). Nontheless there are other mathematical structures that contain infinitesimals. See for example Hyperreal number. --SirJective 15:47, 5 August 2005 (UTC)

To say that a small number x is classed as infinitesimal if the inequality

  |x|+...+|x| < 1

always holds, no matter how large the finite number of n terms in this sum, is absolutely rubbish. Why? If x is greater than zero, then there does exist a finite number of terms n such that the sum is equal to or exceeds 1. No such inequality is possible or even remotely realistic. Anonymous 5:30PM Sep. 2005

Again, anonymous, you're missing the point a little bit. For x in the real numbers, you're absolutely right, as the article states (nothing can be a real number and an infinitesimal at the same time). But it can be useful to think of systems where infinitesimals do take part, as in Nonstandard analysis. The idea is that infinitesimals are not the numbers you're familiar with, and do satisfy that inequality, similarly to how imaginary numbers are not real numbers, and do satisfy the inequality x^2 < 0. Andrew Rodland 09:45, 20 November 2005 (UTC)

Oh rubbish! Nonstandard analysis is even a bigger load of hogwash than standard analysis.71.248.131.74 02:55, 6 February 2006 (UTC)

You don't have to get angry with people when you don't understand the explanation. If you have ever attended even a reletively informal lecture on analysis, it would be clear why infinitesimals are an important concept. Take a math class, ask polite questions, or make a cogent counter-argument. Don't pawn your unwillingness to attempt an understanding outside your own little sphere off on other people.66.153.117.118 20:04, 26 February 2006 (UTC)

Here is a very simple example of infinitesimals in an ordered field, comprehensible to any secondary-school pupil. The field contains all rational functions with real coefficients. A rational function is a polynomial over another polyomial. To make this an ordered field, one must assign an ordering compatible with the addition and multiplication operations. Now f > g if and only if f − g > 0, so we only have to say which rational functions are considered positive. Write the rational function in the form of a polynomial plus a remainder over the denominator, where the degree of the remainder is less that the degree of the denominator. Call it positive if either (1) the leading coefficient of the polynomial part is positive, or (2) if polynomial part is zero, the leading coefficient of the remainder is positive (after, if necessary, multiplying the top and bottom of the fractional part by −1 to make the leading coefficient of the bottom part positive).

By this definition, the rational function 1/x is positive but less than the rational function 1. Now if you add

(1/x) + ... + (1/x)

with n terms in the sum, you get n/x, which is positive but still less than 1, no matter how big n is.

I leave it as an exercise to show that this ordering is compatible with the addition and multiplication operations, so that this is indeed an ordered field. Thus 1/x is an infinitesimal in this field.

So, Mr. or Ms. 71.248.131.74, you are mistaken. Michael Hardy 01:10, 27 February 2006 (UTC)

This is such a good example, Michael, I'm going to put it in the article! --Toby Bartels 01:12, 8 April 2006 (UTC)

I've always had a problem with the use of the term "rational function." It implies a function that maps rational numbers onto the rational numbers. The constant function f(x) = pi may be a "rational function, but it is hardly rational! Phrases like "polynomial ratio" or "polynomial quotient" are more self-explanitory.

If the nonarchimedean field of polynomial ratios is modified to define arbitrary relations among such functions to have the same truth as the corresponding relations for all but a finite number of natural values of the variable, the field becomes a model of a hyperreal number system, with all first-order statements about polynomial ratios having the same truth value as the corresponding statement about standard or real numbers. If one adds this coment to the example here, the same section could be added to the article on hyperreals. Alan R. Fisher 21:24, 22 November 2006 (UTC)

Problem with the definition

From the article:

If x is a positive number (or positive element of any ordered algebraic structure), then x is infinitesimal if there exists a positive element y such that, for every natural number n, the multiple nx is less than y. That is, the inequality

$|x|+\cdots +|x|<y\!$

Now assume that the structure in question is an ordered field with z such that 0 < z < 1/n for all natural numbers n (for example, in hyperreals there is such a z). Let x be a non-zero element of the field. Let y = |x|/z. Since z is infinitesimal, 1/z is infinitely large, and hence $|x|+\cdots +|x|<y$ for any finite number of additions. Hence, all elements of the field are infinitesimal.

Is this really the intended definition? The definition seems to give the desired notion if one replaces y with 1.Punainen Nörtti 10:34, 9 June 2006 (UTC)

This is what we need as a definition of Archimedean (if that is to mean: there are no infinitesimals). However, you're right that this is incorrect as a definition of infinitesimal. The problem is that, without an absolute scale (provided by the multiplicative unit 1), there is no absolute meaning of infinitesimal (nor infinite). If you take y = 1, then you get the correct definition of infinitesimal; but this only works if there is such a thing as 1 in the algebraic system.

The real problem is that the idea "Archimedean" = "there are no infinitesimals" is wrong. It is true in fields, but false in rings, and meaningless in groups. (A ring with no infinitesimals is still non-Archimedean if it has infinite elements.) So I've fixed the definition to use only relative notions of infinitesimal (and infinite). But it might be better to rewrite the definition to avoid those terms entirely.

—Toby Bartels 22:54, 20 September 2006 (UTC)

Right now it seems to have the same problem, i.e. we're restricting to algebras with a notion of "size", but then we're saying x is infinite(simal) if it is infinite(simal) with respect to *every* positive element, but of course this can never be the case, since if x is infinite with respect to some positive y, x>y, so x is positive, but x can't be infinite relative to itself. I've rewritten it by restricting to algebras which contain 1, and defining infinite(simal) as infinite(simal) relative to 1.

--23:19, 19 July 2008 (UTC)

Capitalized or not?

Should it be "Archimedean" or "archimedean"? Note that archimedean field uses the downcase version. Oleg Alexandrov 01:25, 13 Feb 2005 (UTC)

I'm going to post this to WP:MATH. - grubber 16:02, 18 April 2007 (UTC)

Clean-up

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)

Archimedean property in constructive analysis

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)

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)

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)

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)

Proof revision

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)

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)

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)

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)

Well, I don't exactly share your views on these articles... but you're welcome! Melchoir (talk) 00:30, 8 August 2008 (UTC)

Clarification needed

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)

I have added some more explanation. Is this what you had in mind? siℓℓy rabbit (talk) 13:16, 9 December 2008 (UTC)

Yes! Thank you. -- The Anome (talk) 18:01, 9 December 2008 (UTC)

Definitional problems

This article's definition of "x is infinitesimal with respect to y" is flawed. In the real number system, which is defined to be Archimedean, let x=-1 and y=1. The definition falls apart if we assume that n is a natural number. If we don't make such an assumption, what's preventing n from being, say, y/x (in a Nonarchimedean system)? I think absolute value symbols might be useful here. On a related note, if you don't use them, how can this be applied to the complex numbers (which are unordered)? --Thinboy00 @214, i.e. 04:08, 5 February 2009 (UTC)

Is this the definition you're referring to?

Let x and y be positive elements of a linearly ordered group G. Then x is infinitesimal with respect to y (or equivalently, y is infinite with respect to x) if, for every natural number n, the multiple nx is less than y, that is, the following inequality holds:

$\underbrace {x+\cdots +x} _{n{\text{ terms}}}<y.$

[1] Melchoir (talk) 04:55, 5 February 2009 (UTC)

Any objections to this addition?

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)

Archimedean axiom added

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)

I removed your symbolic additions. They did not accurately describe your cited, English language addition. Cliff (talk) 21:52, 27 July 2011 (UTC)

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)

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)

Definition removed from lead. Cliff (talk) 12:55, 22 August 2011 (UTC)

Archimedean Principle

Archimedean principle redirects here. It should perhaps redirect to Archimedes' principle, a totally different thing.--Mongreilf (talk) 21:28, 14 November 2011 (UTC)

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)

disambig on Archimedian rinciple then?--Mongreilf (talk) 22:39, 14 November 2011 (UTC)

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)

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)

This Wikipedia entry is completely irrelevant

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)

Properties

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)