Talk:Infinitesimal

message: I thought about your comment regarding circularity and it occurred to me that you are getting confused. So I will try to help you understand this. Gradient = rise/run.   Rise = f(x+n)-f(x-m)  Run = m+n  So, gradient k = f(x+n)-f(x-m)/(m+n)  => f(x+n)-f(x-m) = k(m+n). We don't know k but we know both rise and run so we can find k. To convince yourself there is no "infinitesimal remainder", divide both sides of f(x+n)-f(x-m) = k(m+n) by (m+n). On the left you have what you started out with and on the right you have k. For any difference quotient, you work with the left hand side so that after cancellation you will have one term without any m or n in it. This term denotes the gradient when m=n=0, that is, both distances on the side of the tangent at the tangent point are zero. Study diagrams in files to get a better understanding. Now, if you wish to find k for any one of the other secant lines that are parallel to the tangent, then you must know their (m,n) pairs. Finding a relationship between m and n helps. However, k will be the same for all the secant lines that are parallel to the tangent.  BTW: The terms in m and n are not remainders! But their sum is always zero because the secants are parallel to the tangent. Each secant has its own (m,n) pair which makes all these terms zero. For example, consider f(x)=x^2. f'(x)=2x+(n-m).  This is exactly the derivative. 2x+(n-m)=2x always.  If x=1, then all the following are valid gradients: 2(1)+(0-0);  2(1)+(0.005-0.005); 2(1)+(3-3); 2(1)+(m-n) Note that m=n in the case of the parabola. This is not always true for every function. In fact, it's hardly ever true for most other functions. 12.176.152.194 (talk) 17:05, 1 January 2012 (UTC)
replies: text: You refer to "terms".  I assume therefore that you are working with polynomials.  The technique you outlined is interesting, but it seems to be what Pierre de Fermat did a few decades before Newton and Leibniz, in developing his method of adequality.  If you figured this out on your own that's certainly brilliant.  But, believe me, calculus has come a long way since then.  In particular, dealing with "terms" can only be done in the context of polynomials.  Alternatively, you need power series which would allow you to account for analytic functions.  Already in power series you would need to delete the infinitesimal remainder at the end of the calculations.  Furthermore, to apply this to functions that are not analytic, you would need the usual differential quotient and the standard part function.  Tkuvho (talk) 17:14, 1 January 2012 (UTC)
replies: text: Not at all the same. It does not matter what you are working with, polynomials or any other function. If the function is smooth and continuous, it will work. No, Fermat had no idea about this. In fact no mathematician before me knew any of it. And although I am trying to encourage you to study it, there is some learning to be done. If your neurons are firing connections with any previous mathematics, you are not getting it. There are no remainders - infinitesimal or otherwise. I hope you will somehow understand this. It seems to me that you have a big stumbling block where this is concerned. There are no infinitesimals. Not in theory and not in reality. 12.176.152.194 (talk) 17:17, 1 January 2012 (UTC)
replies: text: A curve ball for you - the only objects we know about are the rational numbers and incommensurable magnitudes. Nothing has changed since Archimedes. Most of what you learned in real analysis is either unsound or just plain wrong. Usually the latter is true. 12.176.152.194 (talk) 17:26, 1 January 2012 (UTC)
replies: text: You will find some here and very well explained, too.  Tkuvho (talk) 17:26, 1 January 2012 (UTC)
replies: text: Has lots of misconceptions and errors. Not much different from anything else like it. Are you Keisler?12.176.152.194 (talk) 17:29, 1 January 2012 (UTC)
thread_title:  Tkuvho Responses 3 

message: An important step in learning the New Calculus is first realizing where the standard calculus is wrong. You cannot divide by h ever in the standard difference ratio. You can divide by (m+n) always  in the New Calculus. Why? Every term of the numerator f(x+n)-f(x-m) contains a factor of (m+n). After cancellation (taking the quotient), exactly one term will be the gradient of the tangent line [distance pair (0;0)]. To find the gradients of all the parallel secants we use the terms in m and n if we want to be "devout". However, there is no need to do this because their gradients are all equal to the tangent gradient. Now we can find distance pairs in (m,n) for other reasons and there are many interesting reasons - especially in the theory of differential equations which I have researched using the new calculus. So, what you have to do is forget everything you learned and interpret what you read literally. It will take a while even if you are extremely smart. I have found that it's much easier to teach someone who has not learned standard calculus. 12.176.152.194 (talk) 19:47, 1 January 2012 (UTC)
thread_title:  Understanding Cauchy's Kludge 

message: This discussion is not about my New Calculus. Now, although I do not mind whether you mention my New Calculus or not, I will mind if you mention it without proper attribution (my name and web page). I will win any argument in a court of law if it comes to this. Not threatening, just warning. Cauchy's kludge, secant method, distance pairs, etc are also my copyright phrases, not to be mentioned without correct attribution. What I have noticed about academics is that they are cynical until they understand and then they think it's no big deal. Well, it is a big deal because I was the first to think of it. It is also a big deal that I have corrected three great mathematicians: Newton, Leibniz and Cauchy.  Although I can't stop you from quoting my work with the correct attribution, I would prefer that you do not quote my work at all. 12.176.152.194 (talk) 19:05, 1 January 2012 (UTC)
replies: text: I do believe that is a legal threat.  You can stop us quoting your work if it's under copyright, except for "fair use", which our discussion trying to find out whether it has any possible validity seems to fall under.  If you don't want it discussed, you shouldn't have brought it up.  — Arthur Rubin  (talk) 02:11, 2 January 2012 (UTC)
replies: text: I can stop you quoting it if you do not use correct attribution and if you do quote it without correct attribution, I will stop you.  Look Rubin, you annoy me intensely. I have repeatedly informed you that the original discussion was regarding infinitesimals. You kept coming back to my New Calculus. Thukvo continued to ask me about it and the dialogue is primarily with Thukvo, not you. I do not agree with most of your views because they are wrong. If you think this is a threat, that's your problem. 12.176.152.194 (talk) 04:57, 2 January 2012 (UTC)

text: Self-promotion? Have you read any of my comments? I have been trying to contribute to this article. It is full of non-factual statements. Although Thukvo asks me questions regarding my calculus, I keep returning to the main topic. I am prepared to stop right here if Thukvo ceases to ask me questions. I even recommended he contact me via private email if he wishes to continue the discussion. Rubin is an annoying trouble maker with a lot of time on his hands. As for self-promotion, what do you call Rubin's page on Wikipedia? It has nothing notable or remarkable.12.176.152.194 (talk) 05:00, 2 January 2012 (UTC)
replies: text: The only non-factual statements made are by you, and possibly those attempting to interpret your <redacted> "New Calculus".  — Arthur Rubin  (talk) 05:38, 2 January 2012 (UTC)
replies: text: You are the main reason I would prefer no references are made to my work. Your previous <redacted> definition is your wrong interpretation of my definition. 12.176.152.194 (talk) 14:00, 2 January 2012 (UTC)

text: Agreed. The relevant OR is Cauchy's Kludge. I will not respond to any more questions on my New Calculus or any mathematics not related to this topic. 12.176.152.194 (talk) 14:00, 2 January 2012 (UTC)
replies: text: "Cauchy's Kludge" (the name, that it is a "kludge", and the alleged error in Cauchy's work) is also your original research.  — Arthur Rubin  (talk) 15:03, 2 January 2012 (UTC)
replies: text: And so? I am not disputing this fact. iNic asked me what OR and I responded. What's your problem Rubin? Perhaps a new pair of reading glasses is in order? Still living with your mother? 12.176.152.194 (talk) 15:20, 2 January 2012 (UTC)

text: Good arguments I have. Patience for Rubin I do not. Rubin and I go back a long way. There is no love lost between us. I can assure you he probably dislikes me more than I dislike him. What do you say about the tone of Rubin's comments? Do you think he is not attacking me? He is very disdainful and continues to accuse me falsely. A normal person experiences what's called annoyance. 12.176.152.194 (talk) 15:44, 2 January 2012 (UTC)
replies: text: Yes, it is a kludge and it feels good to see you squirming because just about everything you think is knowledge is based on this Kludge. I almost feel sorry for you Rubin. Let's see - Einstein proved to be wrong. Next to follow will be your fake hero Abraham Robinson? 12.176.152.194 (talk) 15:23, 2 January 2012 (UTC)

text: Don't tell me you suffer from reading problems also? Sorry, I meant to say he has been proved wrong. 12.176.152.194 (talk) 15:41, 2 January 2012 (UTC)
replies: text: Have you been following the news lately? I think we must discuss only the topic here - this article. Practice what you preach! 12.176.152.194 (talk) 18:14, 2 January 2012 (UTC)
thread_title:  A word of warning 

message: My work has been published online. That I have a website means it is copyrighted. Furthermore, it is dated so no one can say it's not original. Don't give me that nonsense regarding your knowledge of legal matters. One more thing - I did not bring up the topic, I have been asked several questions and referred those readers to the material. They did not have to read it or continue to ask me further questions.  12.176.152.194 (talk) 05:03, 2 January 2012 (UTC)
replies: text: You did bring it up, because it is the only source you have given for the assertion that infinitesimals are even problematic.  I still don't know what you have against R(((ε)))).  — Arthur Rubin  (talk) 05:36, 2 January 2012 (UTC)
replies: text: Rubbish. I did not bring it up. I was asserting that the modern theory of infinitesimals started with Cauchy's Kludge. Therefore in this respect, the file I referred to (Cauchy's Kludge on my web site) is indeed not only relevant but central to the discussion. There are other sources, many of which are published online, not necessarily also published in the form of a physical book. I have already explained that R(((ε)))) is a figment of your troubled imagination. We had this discussion years ago and neither you nor Michael Hardy could not see the light then. What makes you think you will see it now? The entire theory is absolute rot. But I can't argue against this because Abraham Robinson, bless his dead little Jewish heart, actually got to print his wrong ideas. 12.176.152.194 (talk) 14:07, 2 January 2012 (UTC)

text: I have spent sufficient time satisfying myself that Robinson was an idiot. As for wasting time, it's not really a waste of time pointing out that your article contains many non-factual statements regarding Archimedes and infinitesimals. These have been changed but are still not correct. I don't really care one way or the other so no need to respond to me again. If you want to have the last word, be my guest. 12.176.152.194 (talk) 15:48, 2 January 2012 (UTC)
thread_title: Rubin - Legal Threats 

message: It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods. This is outright false. The reference is subject to opinion and debate. Khodr Shamseddine, "Analysis on the Levi-Civia Field: A Brief Overview," http://www.uwec.edu/surepam/media/RS-Overview.pdf
12.176.152.194 (talk) 23:15, 2 January 2012 (UTC)
thread_title:  Another non-factual and misleading claim in the article 

message: The lede was recently shortened in a drastic way.  Wiki policy allows for the statutory 4 paragraphs.  Is there any reason to make the lede much shorter than provided by policy?  Tkuvho (talk) 16:18, 8 January 2012 (UTC)
thread_title:  lede 

message: I removed the last sentence from this paragraph on Levi Field - "It has applications to numerical differentiation in cases that are intractable by symbolic differentiation or finite-difference methods."  It is a matter of opinion. The stated reference (8) does not provide any evidence this is true. 166.249.134.226 (talk) 17:51, 17 June 2012 (UTC)
thread_title:  Subject to opinion 

message: I clarified the nonstandard reinterpretation of "0.999..." that is being taught, with this edit.. And I started a new subsection regarding the Norton and Baldwin reference that led to this discussion.  -Modocc (talk) 21:12, 5 May 2013 (UTC)
thread_title: Partially resolved

message: I presume some legitimate points are being made in this section, but it is such a muddle.
Perhaps the subject head should be "elementary properties". The term "elementary" is introduced
here, and it is true that the Archimedean property is a consequence of the completeness
property of the real numbers, and stating the completeness property as the LUB property
does involve making statements about sets of real numbers. So it evidently is not an elementary
property of the real numbers.
The section also refers repeatedly to first-order logic (FOL), and that somehow non-elementary properties
cannot be stated in FOL.   The section
states that logic with quantification restricted to elements and not sets "is referred to as first-order logic".
But FOL is not only compatible with set theory, but widely used in combination with it throughout mathematics;
The second paragraph seems to contradict itself, and I do not have the energy to try to sort it out.
It states that the Archimedean property can be expressed by quantification over sets. The
LUB property is indeed expressed using sets, but the Archimedean property can be expressed
as:
∀x, y ∈ R. x > 0 ∧ y > 0 ⇒ ∃ n ∈ N. n ⋅ x > y
and this does not involve quantification over sets.
The second paragraph then goes on to make further points, but I am unable to follow
the argument.  — Preceding unsigned comment added by Crisperdue (talk • contribs) 20:09, 12 June 2013 (UTC)
replies: text: Actually, the Archimedean property is in L_{\omega_1 \omega}, as the quantification over N is in the metalanguage. (see Infinitary logic#Definition of Hilbert-type infinitary logics for notation.)  You'll note that hyperreals satisfy the displayed definition of the Archimedean property, with N being the nonnegative hyperintegers.  — Arthur Rubin  (talk) 23:05, 12 June 2013 (UTC)

text: I spent some time digging through the history of this section last night.  In case anyone's interested in what went wrong with this section, here's the deal.  It started out as a completely conventional exposition of first-order infinitesimal theory, just like I learned it in school.  In particular, it asserted that not all properties of the hyperreals are identical to the reals (after all, they'd be kinda useless if they were totally identical to the reals, eh?), and it asserted that, since the reals are the only complete ordered field, we can't expect the hyperreals to be complete.  An IP address then came in and added some rather angry (my description) and ungrammatical (objective judgement) edits to the section to assert that one can extend the reals in a way that preserves all properties, and also added the last sentence of paragraph 2 that begins "This is also wrong..." which contradicts the sentence just before it.  Some long-suffering editor came in afterwards and cleaned up the grammar and made things read a little better, but the last sentence in the paragraph is still standing there as a direct (and clearly intentional) contradiction of the earlier assertions.  I'd try to fix it up but unfortunately my knowledge of this subject is too soft for me to be sure what's correct here.  I know that simple expositions of the hyperreals carry over only first order properties.  I also know that Robinson spent what seemed to me to be a large (and difficult) amount of space in his book working to carry forward the second order properties of the reals as well (at least, I think that's what he was doing), which would seem to imply he was striving for a version of the hyperreals which were complete.  That agrees with some things I've read elsewhere on the web.  Presumably, the catch is that from inside the hyperreals, you can't tell the difference between a real and a hyperreal, so you can't form, say, the set of all values infinitesimally close to 1, which obviously doesn't have either a GLB or LUB. Salaw (talk) 15:41, 11 February 2014 (UTC)
replies: text: Thanks for your interest. The IP is probably influenced by the internal set theory viewpoint, where they really are identical to the reals (in fact, they are the reals). I remember a few years ago there was an editor making mistakes of this sort that had to be corrected by other users. He apparently moonlighted as an IP without anybody noticing it. If the last sentence is incorrect the simplest thing would be to remove it. Tkuvho (talk) 15:56, 11 February 2014 (UTC)
replies: text: If you check the French wiki page for non-standard analysis you will notice that it is dominated by the IST outlook.  Checking through its history may help identify the editor but who cares? Tkuvho (talk) 15:57, 11 February 2014 (UTC)
replies: text: By the way, the IST viewpoint is fabulous, but obviously it should not be confused with the NSA viewpoint. Tkuvho (talk) 16:01, 11 February 2014 (UTC)
thread_title:  First-order properties 

message: Perhaps this is not a good question to ask here, but given the very small set of published calculus books using Robinson's methods, I'm curious as to why Henle & Kleinberg's Infinitesimal Calculus isn't mentioned.  I'm no doubt biased, since, as a college student some decades ago, I was totally entranced by Kleinberg's lectures on the subject.  None the less, though perhaps not a complete calculus course, the text seems like a pretty nice exposition of the use of hyperreals in elementary calculus.  Is the issue that its circulation was too small to make it notable, or that it's not widely enough used -- or was it just an oversight? Salaw (talk) 05:58, 11 February 2014 (UTC)
replies: text: Merely an oversight.  Please go ahead and add this! Tkuvho (talk) 13:16, 11 February 2014 (UTC)
replies: text: OK -- I'll have to think a little more about what to say, tho, and find some time to say it.  May be a couple more days.BTW, I should say thanks for the reference to Keisler -- it looks good, and I may end up using it.  We're doing homeschool calculus this semester and I've been finding that, when it comes down to actually using a book to teach the subject, I don't like any of the four or five calculus texts I've got lying around the house.  And writing up my own notes, instead of working from a book, is not only proving (very) time consuming, but I'm afraid that, in the end, it's unlikely to produce something better than a published textbook (no matter what my overwhelming hubris may be telling me).  Salaw (talk) 05:53, 13 February 2014 (UTC)
replies: text: After explaining the basic concepts in terms of infinitesimals, Keisler also provides an explanation in terms of epsilon, deltas. This difficult approach is easier for the students to understand once they have a grasp of the notions already. If they will continue at the university at some point they should be acquainted with the epsilon, delta approach as well (unless the epsilon, delta approach is dropped by the universities in favor of the infinitesimal approach by the time they go to college). Tkuvho (talk) 12:22, 13 February 2014 (UTC)
replies: text: It would be cool if that happened, but then, 30 years ago we were expecting it any day now.  On the other hand, if Keisler's third edition can be taken as evidence, the double standard which made it so hard to use the infinitesimal approach may be fading away.  Unless I've overlooked it in leafing through the book, he nowhere shows the construction of the hyperreals.  Long ago, I asked a math teacher about using hyperreals to teach introductory calculus, and his response was that it was a great idea except that you had to spend the first three months of the course developing the hyperreals.  That's what I mean by a "double standard" -- in algebra, in calculus, they teach the reals by introducing the axioms and taking it for granted that they're consistent and we can find a model for them.  How many elementary calculus students even know what a "cut" is, or have heard of the Peano integers, or would know what it meant to form equivalence classes of all equal fractions?  For that matter, what fraction of analysis students have actually read Landau's Foundations?  Yet there seemed to be a feeling that to use the hyperreals, it was necessary to first show how to build a model for them -- upward lowenheim-skolem, ultrafilters, uncountable symbol sets, the works.  Oh, well, rant rant rant.  Time to go do something useful.  Salaw (talk) 18:28, 13 February 2014 (UTC)
replies: text: My comment about the universities was tongue-in-cheek. Meanwhile I do use Keisler in teaching. As far as the "double standard" is concerned, one of the reasons may be an insufficient awareness of just how much of an idealization the "ordinary reals" really are. But that's a vast subject... Tkuvho (talk) 08:40, 14 February 2014 (UTC)
thread_title:  Infinitesimals in teaching 

message: Its a concern when an article has glaring issues right from the start. Such as the first sentence which gives one the impression that this a Physics article and not a math article.
The use of the word "object" conveys the notion that infinitesimals are a concept that
has some connection with something tangible (ie such that they would be something physical one would find discussed in a physics book)...which they are not. Likewise the discussion of their "measurability" as "objects" is rather strange as nowhere in history has a physics experiment
ever been performed in order to test the hypothesis that one will find something that one cannot measure due to it not having a finite size.  The proper word if one is trying to find
some symmetry here between the math and the physics would be "singularity".  Just saying this because I may want to make some changes here in the interest of confluence if the article above
gets approved. YWA2014 (talk) 04:12, 11 July 2014 (UTC)
thread_title:  Improvements and Article Quality Concerns 

message: The new subsection created by User Prodigy uses the term "infinitesimal" in a different sense from the rest of this page, namely as a function that tends to zero. In particular the expression \lim \frac{\alpha_1}{\alpha_2}=1 is meaningless.  This could perhaps be interpreted in terms of the standard part function but the current version is confusing. Tkuvho (talk) 09:58, 16 December 2014 (UTC)
replies: text: equivalent infinitesimal are two variables whose limits are both zero at the same point and they satisfy \lim \frac{\alpha_1}{\alpha_2}=1. For example, \lim_{x \to 0} \frac{\sin x}{x} = 1. Their main use is to determine the indeterminate form 0/0, which can be helpful when using L'Hopital's rule is complicated. In fact, I also edit Indeterminate form to reflect the change.Degenerate prodigy (talk) 15:27, 16 December 2014 (UTC)
replies: text: Prodigy, all this is true, but this article does not use the term "infinitesimal" in the sense of a variable tending to zero. The viewpoint taken in this article is the viewpoint of Leibniz, Euler, and many others that an infinitesimal is a number that's smaller than every positive assignable (ordinary) number. Therefore there is a problem with using the term in a different sense. Tkuvho (talk) 08:32, 17 December 2014 (UTC)
thread_title:  equivalent infinitesimals 

message: While Isaac Newton and Gottfried Leibniz mostly thought in terms of infinitesimals, they never put the notion on a rigorous footing and in fact sometimes used what is recognizable as a precursor to Karl Weierstrass's more modern concept of the (ε, δ)-definition of limit (aka Epsilontics).
It was only when Abraham Robinson introduced Non-standard analysis that it became possible to build calculus rigorously from infinitesimals, and that approach is not dominant.
As a side note, the reference to Archimedes does not belong in the articles, since neither the method of exhaustion nor the method of indivisibles involves the notion of infinitesimals.
Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:17, 19 December 2014 (UTC)
replies: text: Hi User:Chatul. There is a problem here with the link infinitesimal calculus.  The page used to be an explanation that the term "infinitesimal calculus" meant something different historically from its common meaning today.  The page has been redirected since to calculus. Perhaps the passage should be rephrased to emphasize that infinitesimals were a basic components in the procedures of the infinitesimal calculus as developed by Leibniz. Tkuvho (talk) 20:26, 20 December 2014 (UTC)

text: I see you have a joint publication with W. A. J. Luxemburg. Congratulations! Tkuvho (talk) 20:38, 20 December 2014 (UTC)
thread_title:  Infinitesimals are '''not'''  ''a basic building block of [[calculus|infinitesimal calculus]]'' 

message: While certainly interesting, this is just a single item of mathematical notation that might just as well be described in an article about the concept it denotes. — Keφr 14:27, 26 March 2015 (UTC)
replies: text: If this is merged it may be better to merge to John Wallis which page is more specific than infinitesimal. Tkuvho (talk) 11:07, 30 March 2015 (UTC)
replies: text:  Well, the fact that John Wallis invented the symbol is so compact (so to speak), that it can be probably repeated both in this article, in his biography, and even in the article about the lemniscate symbol, without causing much inconvenience. — Keφr 13:36, 30 March 2015 (UTC)
thread_title:  Proposed merge with [[1/∞]] 

message: The article gives an example but doesn't really define the term. Equinox (talk) 06:44, 19 January 2016 (UTC)
replies: text: It's (supposedly) defined in the reference; as the reference may or may not be a reliable source, and I didn't check, I need to say "supposedly".  The question of whether this is definition has any importance is unclear.  — Arthur Rubin  (talk) 11:04, 19 January 2016 (UTC)
thread_title:  How is "transseries" defined? 

message: Last sentence reads: "Since the background logic is intuitionistic logic, it is not immediately clear how to classify this system with regard to classes 1, 2, and 3. Intuitionistic analogues of these classes would have to be developed first."
Might be helpful to less informed readers (like me :), to flesh out a little what "1,2, and 3" are in this context. Just a few words would probably suffice.  (BTW, I did  try to figure it out from the Smooth infinitesimal analysis main page as well as that of the Intuitionistic logic page to no avail. Granted, some deeper knowledge of such subjects is required of the reader, but it's relatively inexpensive to add just a bit more meat to such bare bones descriptions.) thx  — Preceding unsigned comment added by 2602:306:CF8C:98D0:0:0:0:3E8 (talk) 04:05, 27 August 2016 (UTC)
thread_title:  Section 'Smooth infinitesimal analysis' needs just a bit more? 

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thread_title:  External links modified 

message: I came to this page to figure out the correct mathematical symbol to represent an infinitesimal. I think it's small-capital-Greek-letter-Epsilon. But the page didn't actually answer my question! Would a short section titled "Notation" be helpful and appropriate?  — Preceding unsigned comment added by 73.12.48.32 (talk) 21:15, 19 December 2018 (UTC)
thread_title: Notation

message: An editor has asked for a discussion to address the redirect Smallest number. Please participate in the redirect discussion if you wish to do so.  Steel1943 (talk) 22:06, 20 September 2019 (UTC)
thread_title:  "Smallest number" listed at [[Wikipedia:Redirects for discussion|Redirects for discussion]] 

message: I think most of the information in the lead section, while useful, might do better in the 'history' section; it's very long and there's information in there that doesn't show up elsewhere in the article. Perciv (talk) 18:01, 9 April 2020 (UTC)
thread_title:  Lead Section 

message: "His Archimedean property defines a number x as infinite if it satisfies the conditions |x|>1, |x|>1+1, |x|>1+1+1, ..., and infinitesimal if x≠0 and a similar set of conditions holds for x and the reciprocals of the positive integers."
Which similar set of conditions? It could be interpreted as |x|> 1, |x|> 1/(1+1), |x|> 1(1+1+1),..., or as |x|< 1, |x| < 1/(1+1), |x| < 1/(1+1+1),.... L1ucas (talk) 00:26, 31 August 2021 (UTC)
thread_title:  Archimedean property on infinitesimals is ambiguous 

message: In spite of a comment from 2012 about a "statutory" 4 paragraphs,  I think the lede gets too far into details.   The lede should be a summary. The third paragraph,  for example, seems to be "detailed content" and not summary content.  Maybe also the fourth paragraph.  Why is (or was) there a desire to make the lede as long as possible?  Thanks. David10244 (talk) 13:42, 30 December 2022 (UTC)
thread_title:  Lede too long? 

message: According to the picture, it looks like the infinitesimals are a set of numbers that include zero, but the lead of the article mentions explicitly that an infinitesimal number "is not 0." So, is zero included in the infinitesimals or not? If it is not included, I think this should be indicated in some way in the image, because right now I would say that it is at best misleading. —Kri (talk) 10:41, 9 April 2024 (UTC)
thread_title:  Is zero not an infinitesimal?