Talk:Natural number

WP:NPOV Decimal representations: Elementary education versus mathematics

While elementary education often presents real numbers in terms of their decimal representations, once you try to put things on a rigorous basis they turn out to be cumbersome, and actual mathematics texts use simpler abstract definitions. The article is written from the perspective of elementary education and does not even acknowledge the existence of alternatives. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:14, 29 April 2025 (UTC)

Here, there is no question of point of view. This is only is a question of WP:TECHNICAL: Because of its subject, this article is intended for readers with low mathematical background. Such readers may have heard of real numbers, and generally think of them as infinite decimals. But, probably, they do not care of the distinction between terminating and non-terminating decimals. This distinction and the existence of better mathematical definitions of real numbers are clearly too technical for a sentence that says just that the real numbers extend the natural numbers.

In any case, the lead is not the place for a "rigorous basis" nor for abstract definitions of the real numbers (given in the linked article). D.Lazard (talk) 13:54, 29 April 2025 (UTC)

Would you object to throwing in informally? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:48, 29 April 2025 (UTC)

Not useful, since real numbers can be formally defined by their infinite decimal representation. Moreover introducing "formally" in the lead of this article may confuse many readers who have no idea of the meaning of this jargon term. D.Lazard (talk) 15:15, 29 April 2025 (UTC)

zero "usually" a cardinal/ordinal number

So I want to point out this paper by Harremoës, which as the IP requested does indeed question whether 0 should be considered an ordinal or cardinal number, in fact coming to the conclusion that 0 is not an ordinal number. Specifically the line seems to be "there is no need for the label 'zeroth' in this system because an empty set has no element that should be assigned a label like 'zero' or 'zeroth'". As far as cardinal numbers, I think 0 probably is generally included as a cardinal number (cardinal number says so), but probably if you poked around enough in old set theory textbooks you might find one that uses a convention that the empty set doesn't exist.

So anyway, in terms of the article, the "usually" phrasing definitely seems necessary. But unfortunately, regarding the paper by Harremoës, it is a self-published arXiV paper, so unless there is consensus that he is a subject matter expert, it is probably not reliable enough to cite. You can see his bio, he has a PhD and edited (is still editing?) 3 journals and has given invited talks at conferences and so on, but I don't know if that's enough. Mathnerd314159 (talk) 04:28, 31 May 2025 (UTC)

A self published paper is rarely a reliable source. Anyway, all formal definitions (Peano axioms in particular) define primarily natural numbers as ordinal numbers, and the fact that they can serve also as cardinal numbers is a theorem. So, there is not mathematical distinction between finite cardinal and ordinl numbeers.

About "there is no need for the label 'zeroth' in this system because an empty set has no element that should be assigned a label like 'zero' or 'zeroth'": it would be problematic if mechanical counters could not be initialized to 0. So, at least in common practice, there is a need for 0 as an ordinal number. D.Lazard (talk) 11:09, 24 June 2025 (UTC)

Paper arXiv:1102.0418 would never pass review without some changes. My preference would be to cite both the ISO standard and a few textbooks. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:21, 25 June 2025 (UTC)

Removal of non-paywall link to ISO doc

I was getting ready to revert this change by Physicsworld8, because it removes the direct link to the ISO sample doc, leaving a paywalled version. But then I provisionally changed my mind, in case this is a copyright issue. Of course I think it's terrible behavior on the part of ISO to promulgate allegedly public standards and then charge for access to them, and for that matter I don't like them (ISO) much in the first place and especially don't like them having the arrogance to stick their noses in mathematical usage. However, that's not a question for Wikipedia policy. Or even if it's not a copyright issue, I'm not sure the draft is considered a reliable source.

Frankly a better option would be to find a way to get rid of the ISO cite altogether. Surely we can source this notation to a textbook somewhere? --Trovatore (talk) 08:15, 24 June 2025 (UTC)

My understanding is that iteh.ai is a legitimate licensee of ISO - SIST is a member body of ISO and SIST just reframes the iteh site. Presumably they have gotten permission to show PDF previews, like other providers - they are just more generous previews. In this case it was a happy coincidence that the preview included the relevant info and I didn't have to figure out how to get the full standard. AFAICT it is not a draft, it is a sample of the official standard. It might be possible to get rid of the cite there but the ISO standard is also referenced directly in "Emergence as a term", so that cite isn't going away. Anyways there is WP:PAYWALL which says "Do not reject reliable sources just because they are difficult or costly to access."

As far as the physicsworld8 edit I don't really understand the motivation behind it, even besides deleting the source, it also deletes the chapter/section. This user seems to be a newly-created single-purpose account which modified ISO standard references at 1 edit per 8 minutes for 2 hours. I would say not to think too hard about it and just click that revert button. Mathnerd314159 (talk) 21:24, 24 June 2025 (UTC)

I would prefer to substitute the ISO cite in any case, paywalled or not, because I don't think ISO has any business in mathematics. We should find a real mathematical source. --Trovatore (talk) 21:38, 24 June 2025 (UTC)

Well the sentence is actually about several notations: superscript *, +, subscript >0, >=0, and subscript 0 superscript +. The 1978 (first) ISO standard has superscript *, I would suspect this notation might have originated with ISO. The current ISO standard has superscript * and subscript >0 / >=0, this subscript usage I am not sure where it came from. For the remaining notations I would suspect that the only available sources are notation sections in textbooks that use the notation.

If you don't like the standard, there is a book "Mathematical Expressions" by Jukka K. Korpela that cites the ISO standard and explains how to use the ISO notation with LaTeX and so forth, but personally I think just citing the standard is better.

As far as finding a "real source" that actually discusses the different notations as a subject, good luck - I had enough hardships finding sources for the definition of natural numbers, and even Enderton is a short note. Mathnerd314159 (talk) 22:01, 24 June 2025 (UTC)

Why not both? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:30, 25 June 2025 (UTC)

Limit points?

I question the text This way they can be assigned to the elements of a totally ordered finite set, and also to the elements of any well-ordered countably infinite set without limit points. in § Generalizations. I see no reason to exclude well-ordered countably infinite sets with limit points, e.g., $\omega +1=\omega \cup \{\omega \}$, the second infinite ordinal. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:10, 24 September 2025 (UTC)

It is unclear to what refers "they" in the above quotation. I guess that this refers to the natural numbers used as ordinal numbers in the process of counting. If I am not wrong, the sentence implies that one can count only the elements of an ordered set. In fact, things go the other way: counting establishes a bijection between a set the first natural numbers, and this bijection induces an order on the counted set. It is a remarkable that, for finite sets the result of the counting process does not depend on the order in which the element are counted. This is a nontrivial theorem even if it is not presented to kids this way. For infinite sets, a counting process counts eventually the element of a subset of the given set, which can be or not a proper subset.

So, the section is, at least, confusing or, at most, wrong. The recent edit by TheGrifter80 make things even worse, by inserting pedagogical considerations inside a mathematical content. D.Lazard (talk) 17:01, 24 September 2025 (UTC)

Yes, if they refers to natural numbers rather than to ordinals than it makes sense. BTW, I couldn't find an article on $\omega$, although there is one on $\Omega$. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 11:49, 25 September 2025 (UTC)

Notes

Fundamentals of natural numbers / Intuitive concepts

To me this page could use quite a lot more information on fundamental aspects or properties of natural numbers. Things like counting and ordering, the fact that the natural numbers are infinite, the idea of a successor function, etc. A page like this will have a very broad audience, and most people who come here would not have a background in things like set theory, so these concepts need to be better explained. For example, the first section after History (which is good) is Properties and it immediately jumps into using the successor function to define addition - I would guess many people will be completely lost here. I'm not saying it's wrong, but it is really only understandable if you understand the idea already.

My suggestion is a new section to explain some of these concepts a bit more,.either immediately following History, or as a subsection of Properties. Any thoughts, suggestions, disagreement please? TheGrifter80 (talk) 12:33, 25 September 2025 (UTC)

I would have to say, WP:NOTTEXTBOOK ("Wikipedia is not a textbook"). If someone has no background in set theory, and gets lost trying to understand the definition of addition... well, that is why there are sources. In this case I guess the best source would be Naive Set Theory by Paul Halmos as cited in successor function. Really, Wikipedia is not in the business of making set theory "intelligible to someone who has never thought about set theory before", as Halmos's book does. We do focus on giving "correct and rigorous definitions for basic concepts", but that is where the scope ends - I don't think you or I or any other Wikipedia editor is going to surpass Halmos's book and get this Wikipedia article onto the "list of 173 books essential for undergraduate math libraries". Maybe on Wikibooks, that is a possibility, but that is a separate project. As far as this page, it is about 1 concept, the natural numbers, and other concepts like counting, ordering, countably infinite, successor function, addition, etc. all have their own pages. I might even say this page is too long, a lot of it such as the properties section is unsourced and redundant, but the discussion does help clarify the definition so I haven't pushed for removing it.

As far as my thoughts for improving the article, the easiest thing would just be to move the definition section up, between the "notation" and "properties" sections. Mathnerd314159 (talk) 20:53, 25 September 2025 (UTC)

Agree with a lot of what you've said. I think moving the definition section up would be a good improvement. I suggest this section should include a less formal description as well as the formal definitions. If there are multiple formal definitions of natural numbers that are in some way equivalent, then what is the underlying concept they model?

And agree this is not the place to teach set theory etc. But as you say this is a page on natural numbers, not set theory or peano arithmetic, so when these are introduced I think it's reasonable to provide some context for the reader to understand how and why they fit in. TheGrifter80 (talk) 23:26, 25 September 2025 (UTC)

I have introduced a new section before § History for explaining the intuitive concepts behind natural numbers and how natural numbers have been formalized for modeling these intuitive concepts. It is only a first step for resolving TheGrifter80's concerns. The phrasing of the new section an surely be improved, and the remainder of the article must be updated to refering to this introduction. Also new sections would be useful; for exemple, a section on the infiniteness of the natural numbers. I was tempted to add an explanatory footnote to the new section for linking to Dedekind finite sets, but this link would better fit in a section on infiniteness. D.Lazard (talk) 11:23, 28 September 2025 (UTC)

Thanks D.Lazard for creating the new section Intuitive concept, very good starting point. Some further ideas for discussion and possible extensions of this section.

There are two aspects of number here: "size of a collection" and "rank".

Number as "size of collection" - There is a strong argument to say this is the basic intuitive concept of numbers. For example, Frege (at the start of Foundations of Arithmetic) says the natural numbers give the answer to the question: "how many?". This notion of number is reflected in Hume's principle and the idea of a natural number as the one-to-one mapping between elements in a different sets. Further back, Euclid says: "A number is a multitude composed of units". Important to note: for small numbers "how many" doesn't require counting, it can be apprehended directly. To me, in all of these views cardinality is the starting point of natural numbers - they are a property (the size) of a collection / multitude / set.

Number as a "rank" or "position in a progression" - This seems to be the starting point for axiomatic definitions of natural numbers. Eg Benacerraf: "To be the number 3 is no more and no less than to be preceded by 2, 1, and possibly 0, and to be followed by 4,5, and so forth". Quine: "The condition upon all acceptable explications of number is: any progression—i.e., any infinite series each of whose members has only finitely many precursors—will do nicely."

I'm not suggesting we should be getting into philosophical discussion of "exactly what" natural numbers are in this section (although we could have a section further down discussing some of these ideas). But I do think it is important to clearly distinguish these two aspects or uses, while acknowledging that the intuitive concept of natural number encompasses both of them on an equal footing. TheGrifter80 (talk) 16:53, 28 September 2025 (UTC)

This seems now like it's getting to be a bit overwhelming for the beginning of this article, to the point that we're starting to mislead readers about what natural numbers are for and what they're about. I'd recommend we make the discussion of possible philosophical interpretations a bit more concise (or move the details somewhere deeper down the page) and make sure we add some discussion of what can be done with natural numbers – in particular, arithmetic – closer to the top. –jacobolus (t) 05:26, 7 October 2025 (UTC)

Yes you're right, this early section needs to be made more concise and some of the material can be developed in another section further down if necessary. It might be better with more plain language and straight to the point. Any particular suggestions for changes, and where do you think it's getting misleading? I'll amend, or of course feel free to go ahead. TheGrifter80 (talk) 07:36, 7 October 2025 (UTC)

I don't necessarily mean that anything said there is directly misleading, but more that the focus gives an impression about what natural numbers are about / used for that skews to certain narrow uses and away from their main use, which is representing data and doing calculations. –jacobolus (t) 07:50, 7 October 2025 (UTC)

Besides arithmetic, the other fundamental topic that should be mentioned quite near the start, and probably unpacked more fully in the top half of the article, and not only as "history", is the possible ways of representing natural numbers: in particular, number words, tally marks, counting boards and sliding-bead abaci, various written numeral systems, binary signals in a computer, etc. All of these are more important than e.g. a particular set-theoretic representation. –jacobolus (t) 05:34, 7 October 2025 (UTC)

Lead - opening paragraph

The opening paragraph of the lead is mostly devoted to whether 0 is or isn't a natural number. Do we need this much detail here? I'm not trying to re-open the discussion on whether 0 is or isn't a natural number, but suggest that the article would be improved by making this paragraph more concise, for eg:

In mathematics, the natural numbers are the numbers 1, 2, 3, and so on, and often (but not always) the number 0. Other names for the natural numbers are counting numbers and whole numbers.

If it was necessary we could have short section in the body going into further detail about different views on whether 0 is or isn't a natural and clarify various use of terms (whole, counting, etc). TheGrifter80 (talk) 04:19, 7 October 2025 (UTC)

I rewrote this paragraph, but I left the fact that the other names are less used and sometimes ambiguous.

By the way, I edited the whole lead (one edit per paragraph) for making it simpler, less technical and more accurate. Also, I added a mention of addition and multiplication, which, surprisingly, were not mentioned in the lead before. D.Lazard (talk) 13:35, 7 October 2025 (UTC)

Thanks, that is excellent. TheGrifter80 (talk) 22:45, 7 October 2025 (UTC)

How about moving the previous discussion of alternate names to the "notation" section, maybe renaming it to "Notation and nomenclature"? I don't like how "positive integer" and "non-negative integer" redirect here but don't appear in the article. Also checking ngrams it looks like "positive integers" and "whole numbers" are the two other terms that should be in the lead - "counting numbers" is not very common. Mathnerd314159 (talk) 23:46, 7 October 2025 (UTC)

Google ngrams uses all the books that Google could get their grubby mitts on, which naturally includes irrelevant and/or unreliable sources. It's probably more useful to look at mathematics and mathematics-education literature specifically and see how common terms like counting number are there. Stepwise Continuous Dysfunction (talk) 03:37, 11 October 2025 (UTC)

I reintroduced some of this material as you suggested, please have a look when you can. Feel free to change the name of the section if you think "Nomenclature" is preferable to "terminology". TheGrifter80 (talk) 13:36, 14 October 2025 (UTC)

@Stepwise Continuous Dysfunction: well there is : "it's unlikely you'll have to know any of these terms [natural number, counting number, positive integer] besides positive integer." That is probably the clearest statement possible regarding the primacy of "positive integer" as a term over "counting number" (and over "natural number", but that discussion closed).

A Google Books search of the literature is not so clear on what is most "common", all it can really say is how these terms are used. Roughly what I see is:

• "positive integers", "natural numbers": used in university-level mathematics publications, as well as in secondary education
• "whole numbers": used in secondary education, and also a lot of teacher-oriented "how to teach numbers" type publications. And also in 1900's-era education
• "counting numbers": used in "for dummies" type books, GED, GCSE, GMAT, a few K-5 children's flash cards, then 1900's-era education

So the ngrams result seems reasonable - it doesn't seem like it is particularly biased by irrelevant or unreliable sources. I guess qualitatively I could say that the "for dummies" type books of the "counting numbers" sources seem like low-quality sources, they are tertiary sources often written in a rush with little editing or oversight, but practically per Wikipedia policy they are still considered reliable sources. IDK, I just went with a list of terms. It is the sort of question where Wikipedia is unable to achieve a consensus, and therefore because Wikipedia is founded on principles of decision by consensus it is unable to come to the best decision, and has to settle for a relatively mediocre least common denominator.

@TheGrifter80 I had no complaints about your work, but I had a more substantial revision in mind. Title is fine. Mathnerd314159 (talk) 19:00, 14 October 2025 (UTC)

The page you just linked starts with "Up until now, most of the numbers you've dealt with have been what are called the counting numbers or natural numbers". Frankly this doesn't seem like a particularly great source about terminology; it seems to be a remedial math book aimed at a lay audience looking to brush up on what they learned or didn't learn in school. –jacobolus (t) 19:05, 14 October 2025 (UTC)

Yes, that is the typical quality of the sources that mention "counting numbers". Mathnerd314159 (talk) 16:16, 15 October 2025 (UTC)

My point is, (1) your use of the quotation "it's unlikely you'll have to know any of these terms" is a significant mischaracterization of the source, and (2) this was a pretty questionable choice of source to begin with. –jacobolus (t) 16:43, 15 October 2025 (UTC)

Thanks Mathnerd314159. That does look good. I have one suggestion and interested to see what you and others think. We could have "Terminology and notation" as a brief section immediately following the lead to introduce the various terms, the set notation N and Z, and the two definitions of the naturals. ie Just a short and basic intro to things used throughout the article.

Then further down the article we could then have another section or subsection called "Zero as a natural number" (or something similar) where we can give all the details you have in paras 2 and 3 about conventions in different fields. Also I think it might be useful to provide some discussion of why there are two definitions, in what contexts is it useful to consider 0 a NN or not. Let me know what you think. TheGrifter80 (talk) 07:56, 15 October 2025 (UTC)

I'm not sure what the purpose of such an introductory brief section would be. The lead already introduces that there are two definitions and the typical notation. With your proposal it sounds like we would have this information about set notation and so forth duplicated in three places.

Regarding a separate "zero as a natural number" section, I don't think it is typical to have such sections in Wikipedia articles. It seems that the information is usually distributed among less boldly named sections. I will admit that splitting up the information between the history section and the terminology section and so on is a bit awkward and redundant but it doesn't seem that bad. I think one of the key issues here is that although there are a fair amount of sources on this zero issue, most of them are passing mentions and I would say there is not enough for notability - unless we are really willing to scrape the bottom of the barrel and count these remedial math books and so forth as reliable sources. But as jacobolus says, these are actually pretty bad sources. Mathnerd314159 (talk) 16:33, 15 October 2025 (UTC)

Ok fair enough. My idea was only that "Terminology and notation" could be more about the basic language (terms, notation) used to identify the natural numbers, rather than a

full catalogue of different conventions relating to zero.

I think that info does belong somewhere in the article but to me it's probably more interesting to know why there are these differing perspectives rather than just seeing them listed, and that discussion should not be near the top of the article.

But I can see your point of view and it does it all tie in as you've presented it. TheGrifter80 (talk) 00:24, 16 October 2025 (UTC)

Combinatorics

The introduction currently says, Combinatorics is, roughly speaking, the study of counting methods for sets depending on one or several natural numbers. I am not sure what this is trying to convey. The article does not mention combinatorics again, so nothing that comes later clarifies this remark. Stepwise Continuous Dysfunction (talk) 03:39, 8 October 2025 (UTC)

Sure that there would be no harm to remove this sentence. However it is in a paragraph listing areas of mathematics that are primarily devoted to natural numbers. The question is thus whether we must remove the sentence or planning to add a section about combinatorics. Personally, I would be in favor of the second option. For example, the pigeonhole principle is clearly a property of natural numbers that belongs to combinatorics. Similarly, the inclusion–exclusion principle is a property of natural numbers (viewed as cardinal numbers) that is fundamental in combinatorics. D.Lazard (talk) 10:55, 8 October 2025 (UTC)

I just don't know what the sentence is trying to say. Do the "sets" depend on the "one or several natural numbers"? If so, what does that mean? Do the "counting methods" depend on the "one or several natural numbers"? I know it's meant to be an informal description ("roughly speaking"), but it's so informal that I can't parse it.

If I were trying to give an informal description of combinatorics, I'd say it's where we count the number of objects or patterns of a specific type. Saying what that type is may involve one or more natural numbers (e.g., "How many ways are there to list 5 names?"), but jumping to that skips over the idea about what we're trying to count. Stepwise Continuous Dysfunction (talk) 14:52, 10 October 2025 (UTC)

The set of the partitions of a natural number ⁠$n$⁠ and the symmetric group ⁠$S_{n}$⁠ are basic examples of sets depending on a natural number, and are among the first families of sets studied in combinatorics. Maybe, you know a better way for expressing this? D.Lazard (talk) 15:09, 10 October 2025 (UTC)

What sets don't depend upon a natural number eventually? Even a continuous set will have a dimension or a genus or some such quantity. As written, the phrasing just wasn't conveying any information; I suspect that only people who already know what combinatorics is would have been be able to understand it. I have tried a rewrite. Stepwise Continuous Dysfunction (talk) 03:23, 11 October 2025 (UTC)

Analysis

I looked at a small set of analysis texts and they all either included 0 in the natural numbers or did not define the term. The article cites a single RS for starting with 1. If someone has access to a large collection of analysis texts (at least a dozen) I'd be interested in a headcount of 0 versus 1. Is there an appropriate tag to request more sources? {{Cn}} doesn't seem appropriate for this case. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 10:01, 15 October 2025 (UTC)

I don't think you need a large number of sources. In the western world, zero wasn't "natural" until Fibonacci. Then there are encyclopedias like https://www.britannica.com/science/natural-number, https://enciklopedija.hr/clanak/prirodni-brojevi, https://mathworld.wolfram.com/NaturalNumber.html... Ponor (talk) 12:59, 15 October 2025 (UTC)

A small sample means a large margin of error. Didn't Fibonacci precede the modern conception of analysis by half a century? I'd date it from Weierstrass or later. The encyclopedia articles mention both conventions. `Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:25, 15 October 2025 (UTC)

There's no error, we don't pick sides. Both one and zero ARE used as the first element, unlike 2, 3 etc. Instead of counting the number of RS that support this or that view (which would be our WP:OR), we can rely on WP:TERTIARY sources such as other encyclopedias: "Reliable tertiary sources can help provide broad summaries of topics that involve many primary and secondary sources and may help evaluate due weight, especially when primary or secondary sources contradict each other." Some other valuable sources that start with 1: https://www.treccani.it/enciclopedia/numero_(Enciclopedia-Italiana)/ or https://books.google.com/books?id=DvIJBAAAQBAJ Ponor (talk) 15:35, 15 October 2025 (UTC)

How is In contrast, number theory, analysis, dictionaries, and most schoolbooks (through high-school level) typically define natural numbers as starting at one. not picking sides? It's making four claims about prevalence of one convention versus the other, and each of those claims needs to be verifiable. Citing a small number of textbooks doesn't establish the claim unless one of the textbook itself includes a survey of usage. The number of texts with one usage or the other needs to be large enough for the difference to be statistically significant.

The footnote for analysis shows two texts; I could just as easily cite KEISLER and it would be equally meaningless; what counts is how many use each conventions, and the cited sources don't address that; it requires a statistically significant sample. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:49, 16 October 2025 (UTC)

As for the dictionaries, both of them show both conventions; they don't support a claim of either convention being dominant. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:49, 16 October 2025 (UTC)

well the source for analysis (and most of these "picking sides" claims) is MacTutor (and for the high school / university divide it is Enderton). For example with analysis, the original source would be this usenet post by Gerald Edgar, "I find, as a general rule, with some exceptions, that texts in algebra include 0 and texts in analysis exclude it." IDK, he was a professor of mathematics, and he was quoted so Mactutor is a secondary source, but ultimately it is just his impression. Is it worth including these "impressions" in the article? It's not like it's WP:BLP or WP:MEDRS content. Mathnerd314159 (talk) 04:56, 17 October 2025 (UTC)

It's probably true, but Netnews is not a RS, so be careful with the wording if you include it. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:08, 17 October 2025 (UTC)

We could certainly report on what one math professor writing in 1994 personally found to be "a general rule, with some exceptions" in the sources they were familiar with. I'm not sure it's that helpful to readers though. –jacobolus (t) 16:17, 17 October 2025 (UTC)

My view on this is that we should say there are two conventions and leave it at that. Trying to parse out which convention is more used in what milieu is always going to risk original research / original synthesis, and isn't worth it. What the reader actually needs to know is: Sometimes zero is included and sometimes it's not, and if it matters to your case whether it is or not, then you'd better check. --Trovatore (talk) 19:14, 16 October 2025 (UTC)

Hmmm, I thought we were discussing the latest 10^N edits to the lead. I agree with Trovatore. Unless a source says this convention is used more than that convention in some subfield, the "analysis" in that section will always sound like original research. Ponor (talk) 19:28, 16 October 2025 (UTC)

To make that concrete, my proposal is that the second paragraph of the "Terminology and notation" section (permalink) should simply be removed. --Trovatore (talk) 19:25, 16 October 2025 (UTC)

Either delete it or replace it with something like

The literature uses both conventions, even within the same field; no formal survey has measured the preferences.

with opposing citations in the same field(s). -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:48, 16 October 2025 (UTC)

I went ahead and boldly removed the paragraph. If you're OK with that then I'd suggest we move on. If you really feel there's value to including this language, well, then I suppose we'll have to keep talking about it. --Trovatore (talk) 21:09, 16 October 2025 (UTC)

Agree with the removal. What about the next paragraph that talks about history, empty set, ISO standard etc? I think this could be removed or relocated to history. TheGrifter80 (talk) 22:49, 16 October 2025 (UTC)

Variants of this paragraph have existed for many years. I note you removed similar material 10 years ago. I guess now that the page is protected, nobody else will try to add it, but it still seems like running in circles. If there is an issue with sourcing, fine, that has plagued this article a lot, but it seems to me there is enough there regarding who uses what convention that something can and should be said. Mathnerd314159 (talk) 05:22, 17 October 2025 (UTC)

Well, obviously I disagree. Possibly something could be said, but I don't think anything should be said. --Trovatore (talk) 17:34, 20 October 2025 (UTC)

References

Alternate names in lead

@Jacobolus: The lead currently state The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used.; this is an incomplete list. I added strictly positive, but Jacobolus reverted it. I believe that the list should include all of the common terms in the literature and that it should be moved out of the lead. Also, is counting numbers ever used outside of elementary education? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:01, 20 October 2025 (UTC)

@Chatul The name "strictly positive integer" is less than 1% as common as "positive integer", and strictly positive integer hasn't even been created as a Wikipedia page title. It is unnecessary to add here, distracting, and potentially confusing. The point of the lead paragraph is not to comprehensively survey every term ever used for this topic, but rather to clearly show the most common names that redirect here, in bold for visibility, so that someone who clicks a wikilink or arrives via Wikipedia or web search for e.g. positive integer or counting number is immediately reassured that they have arrived at the right page about the topic they were looking for. If you want to do a more careful comparison of terms, it belongs somewhere other than the second sentence. –jacobolus (t) 16:26, 20 October 2025 (UTC)

To answer your question about the term counting numbers, a cursory search finds it used in a variety of contexts, including not only primary/secondary education but also psychology, anthropology, comparative linguistics, the history of mathematics, philosophy, ... Perhaps you meant to ask how common this term is in higher mathematics? Not very common. But this article must appeal to an audience of mostly laypeople and students. Topics and framing of specific interest to professional mathematicians shouldn't be emphasized near the top. –jacobolus (t) 17:22, 20 October 2025 (UTC)

Also, I would note that "positive integer" is a subphrase of "strictly positive integer". I think it is unlikely that someone would google "strictly positive" by itself and expect that phrase to refer to the natural numbers. For me the meaning in https://www.pls-lab.org/Strictly_positive was the first to come to mind. Mathnerd314159 (talk) 06:01, 21 October 2025 (UTC)

The complex numbers are a strict superset of the reals with i.

re "[the integers] are made by including [0 and negative numbers].

[The rational numbers] add [fractions], and

[the real numbers] add [all infinite decimals].

[Complex numbers] add *[the square root of −1]*.",

Z = N∪{0}∪{x : x is a negative natural}

Q = Z∪{x : x is a fraction}

R = Q∪{x : x is an infinite decimal}

¬(C = R∪{√-1})

what I mean is that saying that the the complex numbers add i, is like saying that the integers add -1. That is kinda true for a reasonable interpretation of add, but it is a different interpretation than all other uses of add in this paragraph. C\R is not {i}; C is not just one element added to the reals, it is the closure of that set under multiplication/addition/division/etc.

I don't know if this necessarily needs changed, and I can't think of a better way to express it, but it bugs me that this paragraph is expressed inconsistently in this way. EktelestesPragmaton (talk) 17:58, 17 December 2025 (UTC)

I tried rewriting that paragraph. Does that help? –jacobolus (t) 19:25, 17 December 2025 (UTC)

It still has some issues. I'd recommend removing it from the lead and adding material in § Generalizations on $\mathbb {Z}$, $\mathbb {Q}$, $\mathbb {R}$, $\mathbb {C}$, $\mathbb {H}$, $\mathbb {O}$. This could include both axiomatic and constructive approaches. The material should note that while there are natural embeddings $\mathbb {N} \mapsto \mathbb {Z} \mapsto \mathbb {Q} \mapsto \mathbb {R} \mapsto \mathbb {C}$, there is no unique embedding $\mathbb {C} \mapsto \mathbb {H}$.

I'd also suggest removing This makes natural numbers foundational for all mathematics. or considerably weakening it and using a source from mathematics rather than primary education. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 21:35, 17 December 2025 (UTC)

I'm not sure I understand what the dispute is here and I'm not incredibly interested in figuring it out. I just want to urge that this material not be expanded to try to solve whatever the problem is supposed to be. It seems reasonable to mention more expansive number systems and how they are built up from the naturals, but just "mention", not "go on and on about". The current level of detail seems fine.

I do have one picky objection, which is formulating the reals in terms of "infinite decimals", which is not at all the best way to think of them, in particular because it suggests there's something special about base 10. That said, I definitely don't want to introduce Dedekind cuts or Cauchy sequences in this article, because that would be "going on and on about" it. I guess I can live with the mention of decimals, but it does slightly rub me the wrong way. --Trovatore (talk) 21:47, 17 December 2025 (UTC)

I wouldn't say there's a "dispute". I just rewrote this paragraph from:

Many number systems are built from the natural numbers and contain them. For example, the integers are made by including 0 and negative numbers. The rational numbers add fractions, and the real numbers add all infinite decimals. Complex numbers add the square root of −1. This makes up natural numbers as foundational for all mathematics.

to

The most common number systems used throughout mathematics are extensions of the natural numbers, and can be formally defined in terms of natural numbers. If the difference of every two natural numbers is considered to be a number, the result is the integers, which include zero and negative numbers. If the quotient of every two integers is considered to be a number, the result is the rational numbers, including fractions. If every infinite decimal is considered to be a number, the result is the real numbers. If every solution of a polynomial equation is considered to be a number, the result is the complex numbers. This makes natural numbers foundational for all mathematics.

with the goal of being more precise and less potentially confusing. Hopefully this isn't "going on and on about" this topic, but it could also plausibly be boiled down to a sentence or two and unpacked in a later section. –jacobolus (t) 01:20, 18 December 2025 (UTC)

Anything more advanced than "infinite decimals" is not going to be usable for an elementary audience in my opinion. Someone who clicks through to real number can hopefully learn as many details as they want. –jacobolus (t) 01:17, 18 December 2025 (UTC)

I do think the problem is by making it "more precise" you also made it much longer, 55 words to 113 words. I would say, revert to the original paragraph in the lead, or even something more concise, and then write a new thing in the generalizations section on the precise details of how the numeric tower is constructed, and what "adding" an element means in this context (it has a precise interpretation in terms of algebraic closure, subrings, etc.). Mathnerd314159 (talk) 03:13, 18 December 2025 (UTC)

Yeah, looking further, the correct notion is embedding, and this was linked before in the lead, but it was deleted in Special:Diff/1315574958 by @D.Lazard. I do not really understand the justification for this, the edit summary is just for the other change. Mathnerd314159 (talk) 03:24, 18 December 2025 (UTC)

A wikilink to Embedding seems completely useless to the intended audience of the lead section of this article. –jacobolus (t) 06:44, 18 December 2025 (UTC)

Well, the issue with the lead is that it must be written at a very approachable level, while also not telling lies to children (literal children, in this context, given the likely readership). So a wikilink is about the cleanest approach I can think of, explaining that the meaning of "embed" here is a lot more rigorous than its casual connotation. Its utility is the usual utility of a wikilink, if someone wants to learn more about what is meant then they can click on it and dive down the rabbit hole. Mathnerd314159 (talk) 17:27, 18 December 2025 (UTC)

The problem is that "including 0 and negative numbers", "add fractions", "add all infinite decimals", and "add the square root of −1" are of different forms, quite confusing (readers will plausibly misunderstand what "add" means), and in the last example misleadingly incorrect if following the implied pattern of the first 3 examples.

The portion in the lead could probably be shortened to something like

The most common number systems used throughout mathematics – the integers, rational numbers, real numbers, and complex numbers – are extensions of the natural numbers, and can be formally defined in terms of natural numbers.

and possibly combined with a different paragraph. Something like my rewritten paragraph (or even a longer elaboration) could be included further down the page. –jacobolus (t) 06:42, 18 December 2025 (UTC)

This seems very convenient for the lead, eexcept that "are extensions" seems unnecessarily technical for the kead, and would better be replaced by "contain".

To editor Mathnerd314159: the reason of my revert is that "This chain of extensions canonically embeds the natural numbers in the other number systems" is is much too technical for the intended audience of this lead. Moreover, I am not sure whether "a chain of extensions that embeds te natural numbers" is defined anywhere. Also, in the linked article there are many different concepts of embeddings, and I have not found a definition that is directly applicable (without advanced mathematical knowledge) to number systems (indeed, an embedding is a structure preserving injection, and to embed the natural numbers in some structure, one must first define the structure of the natural numbers). D.Lazard (talk) 16:43, 19 December 2025 (UTC)

To editor Jacobolus: Yeah, I agree that anything "more advanced" is out of scope for this article, and honestly I think maybe the mention of the reals is already out of scope. If we are going to mention the reals then I suppose we're stuck with "infinite decimals". It's a bit grating but I don't really have a better suggestion. Do we really want to mention the reals? I'm still a little on the fence about that. --Trovatore (talk) 04:46, 18 December 2025 (UTC)

If we do have to mention the reals, do we have to mention them in the lead? -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:12, 18 December 2025 (UTC)

This generalizations section currently has:

These number systems can also be formally defined in terms of natural numbers (though they need not be).

with a note:

The commonly-assumed set-theoretic containment may be obtained by constructing the reals, discarding any earlier constructions, and defining the other sets as subsets of the final construction

Does this risk giving a misleading impression? My initial reading was that it was saying there are ways to define Q or R (for eg) in a general way without starting with N? TheGrifter80 (talk) 04:52, 8 February 2026 (UTC)

It seems clear to me. Is there another way to read it?

What did you mean by (for eg)? "Exempli gratia (e.g.)" is "for example" and does not stand alone, it needs a comma and a list. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:55, 9 February 2026 (UTC)

Perhaps I am missing something. What is the intended meaning of saying that other number systems need not be "defined in terms of natural numbers"?

I don't see how that can be the case. TheGrifter80 (talk) 00:31, 10 February 2026 (UTC)

@TheGrifter80

just to give a possible example, you could define the integers as the free monoid generated by the two element set {1,-1} quotiented by the equality 1+-1=-1+1=0, then define the rationals, reals, complex numbers, etc. from the integers the way you normally would and you never needed to use the naturals. don't know if that is help in answering your question. EktelestesPragmaton (talk) 12:06, 10 February 2026 (UTC)

Thanks for that example, that does help. I'm not too familiar with abstract algebra, but interested to understand how 1 and -1 are defined in the system. I will look around for an introduction, thanks again. TheGrifter80 (talk) 00:17, 11 February 2026 (UTC)

@TheGrifter80 they are just symbols, they could be literally anything you want and it would have the same structure. EktelestesPragmaton (talk) 08:31, 11 February 2026 (UTC)

Article sections and structure

Looking for some input on the structure of this article, particularly order of the sections and possibilities for new sections. Three suggestions to start:

1. Move the History section to the bottom of the article. I think this article would be best starting off with simple concepts, then gradually introducing more technical detail, but the History doesn't really fit into that flow, so suggest moving to the end.

2. New section on "representations of numbers" (as suggested by jacobolus  last year). Including numerals and numeral systems, probably following "Intuitive concept" section.

3. Section on philosophy of natural numbers. Would only be of interest to a small minority of readers so it would be right down the bottom of the article either as short stand-alone section or part of History.

I will start on these three, but please let me know objections and other suggestions. TheGrifter80 (talk) 00:45, 11 February 2026 (UTC)

I'd recommend starting any major project by looking for sources.

I think this article would best start with some summary of elementary arithmetic of natural numbers. It would also be good to discuss divisibility and primes pretty near the top (but not starting with the current "The algebraic structure $(\mathbb {N} ,+)$ is a commutative monoid with identity element 0. It is a free monoid on one generator. This commutative monoid satisfies the cancellation property, so it can be embedded in a group. The smallest group containing the natural numbers is the integers. [...] Analogously, given that addition has been defined, a multiplication operator $\times$ can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns $(\mathbb {N} ^{*},\times )$ into a free commutative monoid with identity element 1; a generator set for this monoid is the set of prime numbers." which is ridiculously over-technical for the likely audience for this page.) –jacobolus (t) 01:58, 11 February 2026 (UTC)

Thanks, and agree about the paragraphs you highlighted, far too technical. TheGrifter80 (talk) 03:19, 11 February 2026 (UTC)

There is not a structure guide in WP:MOSMATH, but in Wikipedia:Manual_of_Style/Computer_science#Structuring_different_kinds_of_articles there is some advice. To pick out some patterns as concrete suggestions:

• put history between after formal statement but before generalizations, as in the theorems/conjectures section
• put "Terminology" after "Intuitive concept", as in "programming constructs"
• put discussion of representations after history (as in "Algorithms and data structures", "classic problems", etc. which put solution/implementation after history)
• philosophy: would go into intuitive concept as far as being easily explainable, otherwise at end

As far as new sections go, the article generally seems complete, as jacobolus says it is really about following the sources. Mathnerd314159 (talk) 00:47, 13 February 2026 (UTC)

thanks Mathnerd314159.

• put history between after formal statement but before generalizations,

Agree, good suggestion.

• put "Terminology" after "Intuitive concept"

I can see the case either way for which of these first should come first.

• philosophy: would go into intuitive concept as far as being easily explainable, otherwise at end

Agree. I previously had some of this material in intuitive concept but it was too dense as it was written for the start of the article. I will re-add at the end, and look for opinions if any can/should be moved into "Intuitive concept". TheGrifter80 (talk) 03:03, 13 February 2026 (UTC)

I think I'd somewhat pare down the history section here, keep it toward the bottom, and direct people toward Number#History instead (or we could even have a separate article History of numbers (currently a redirect); It doesn't seem that important to prominently feature a history about the concept of 'natural number' per se here. Instead I think we should focus first on the internal structure of natural numbers, second on the relation of natural numbers to other types of numbers, and third on the applications of natural numbers (in greater depth than now). –jacobolus (t) 03:11, 13 February 2026 (UTC)

On the History section, I agree it should be pared down. I suggest we get rid of the "ancient roots" subsection (or summarise it very briefly), as this is just a general history of numbers. We could keep a small section of history as relates directly to what we now call "natural numbers", which is a fairly recent distinction and term, so could be of some interest.

As for applications of natural numbers, I agree and thought another section on arithmetic (at a less technical level) would be useful as you suggested above. Currently some of this is covered in "Properties" but in a formal way. Perhaps also move the paragraph on counting out of Intuitive concepts into this new section. TheGrifter80 (talk) 03:35, 13 February 2026 (UTC)

Edits by the grifter 80 recent and 8 October

you set theorist editors want to present set theory infinity like a paradise garden so explain it very verbosely but avoid stating things that are an embarrassment or misstate things so they are more rosy. Sometimes you present your own material as if it's the material of the professional and original set theorists. These are obviously contrary to Wikipedia policy. Also over verbosely presenting set theory infinity but also avoiding other things, as though there's not enough room, is obviously wrong. You do that because then set theory won't be rosy. And you present concepts that can be explained simply but present them more confusingly, presumably since people could figure out the errors of set theory if explained simply.

The present edit the editor tried to come up with his own concept of cardinality, not that of the cantorists, such that it applies to finite and infinite, and equal and not equal, and is simple, one paragraph. They don't use that because it's mathematically invalid. The natural numbers are injective bijective and surjective with the natural numbers, so claiming the natural numbers are not equal cardinality with the natural numbers because things are left over, pairing the natural numbers with their doubles, is invalid.

You would have to say if you cannot prove equal that proves not equal. Obviously invalid but that IS how set theory infinity works. But that is not what the set theorists say because that leaves out greater and they want proof not observation of not able to prove equal.

What the original and current literature says is that using the concept :

That which applies to to finite by proof also applies to the equivalent infinite without proof

But only in some cases, other cases it's 'does not apply'. This is not proof but presumption and since they cannot apply it universally presumably false. So they apply it to infinite (thus also finite) ≤ and ≥ and =, but don't allow for < or >. Likewise claim 1 + infinity equals infinity contrary to the principal, but cardinal exponentiation follows the principal.

Then they use the diagonal argument to prove their proof of (they use =, your using ≤) won't work. But then use lack of proof is proof of opposite (≠). So then using their proof of ≥ plus that to get, by a definition, >.

Likewise with ordinal size they use can prove 1 + infinity equals infinity (that's never ending infinity not as one of you claims infinity with end, in that case the proof doesn't work) , but can't prove infinity plus one equals the same infinity so claim adding a member (the set of ordered natural numbers) to the (nonexistant presumably) end of an endLess set (the natural numbers) produces an infinity with an end and larger, that is lack of proof of same length is proof of greater length.

Now in original set theory size cardinal size and ordinal size are precisely defined, not by proof or axiom but definitions. Now obviously the above (cardinality ) is not the same as actual size but quite different, and the way I explain it it is obviously erroneous. Thus the original set theorists back to cantor nor present the most professional set theorists don't claim cardinality is the same as size, nor only technically not size. Further repeating explaining cardinality instead of linking to the cardinality article is wrong

The same editor produced both edits, recently and October 8

Victor Kosko (talk) 00:16, 2 March 2026 (UTC)

It's not clear what you're trying to say. If you have specific suggestions I'm happy to look at them - there might be something useful to add. TheGrifter80 (talk) 14:29, 3 March 2026 (UTC)

I was clear

You don't want to believe it so your mind doesn't accept it

I explained to you size is not cardinality yet you again put in an edit claiming that. The set theorists did not mistakenly use definitions for Cardinality instead of axioms for Size. Each of the original set theorists including Cantor claimed size is not cardinality and in YouTube video comments set theorists correct that size is not cardinality. Your obviously violating Wikipedia policy. I explained in detail. You also decided to invent your own set theory concept to define cardinality as if you understand better than they do. No. Your paragraph doesn't work since for example the natural numbers are not ONLY in bijection with the natural numbers but also surjection and injection, so injection does not prove >.

Instead they prove bijection cannot be proved to supposedly prove ≠ THEN use injection to prove not > but they claim ≥. Inventing your own set theory is original research and against Wikipedia policy as well as deceiving

It's the set theorests Victor Kosko (talk) 19:57, 3 March 2026 (UTC)

Clear as mud. You explained nothing, and ad hominem arguments only damage your credibility. You simply made a string of unsubstantiated allegations. Even WP:Fringe theories require a RS. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:46, 3 March 2026 (UTC)

Language models are surprisingly good at understanding even otherwise impenetrable comments, here is the list of concrete suggestions the LLM produced:

• "Size of a collection" section: Explicitly distinguish "size" from "cardinality" - clarify that cardinality is a specific mathematical definition of "size" based on one-to-one correspondence (bijection), and there are other definitions of size (e.g. ordinals can be considered sizes) and also that mathematical notions of size may not correspond to our intuitive notion of size. I think some of this is in the cardinality article, but I guess the treatment here is too abbreviated.
• "Position in a sequence" - so the section is right in presenting the idea that an ordinal number is whatever "comes after" a given set/sequence. But it is wrong in that in implying that all ordinal numbers are natural numbers, or that ordinal numbers behave like natural numbers - the simple example is non-commutativity of ordinal addition, 1+ω=ω but ω+1 is itself. So we need to distinguish "using natural numbers to order" (ordinal numerals) from the formal definition of "ordinal number" (which includes natural numbers but also infinite ordinals).
• Over-focus of article on set theory - Victor seems to want a dedicated subsection on Finitism or Intuitionism, and how some mathematicians (like Leopold Kronecker or L.E.J. Brouwer) rejected the idea of "completed" infinities, to present a more nuanced view of the infinite nature of the natural numbers than simple "the natural numbers are a subset of the cardinals/ordinals".

Comparing between Viktor's comment and this list of suggestions, I am not sure this list will make Viktor completely satisfied, but they do seem like reasonable suggestions. Mathnerd314159 (talk) 21:43, 3 March 2026 (UTC)

there are other definitions of size Cardinality gives the same comparisons as the "how many" meaning of size.

e.g. ordinals can be considered sizes Ordinal numbers carry more information than just size, e.g., $\omega$ and $\omega +1$ are distinct ordinals but imply the same cardinality.

But it is wrong in that in implying that all ordinal numbers are natural numbers, or that ordinal numbers behave like natural numbers Where does it imply that?

If you're thinking of A natural number used to describe a position in a sequence is called an ordinal number., I agree that the wording could be clearer.

Victor seems to want a dedicated subsection on Finitism or Intuitionism His prose is so murky that I'm having trouble parsing it. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 22:03, 4 March 2026 (UTC)

'If two collections do not have the same cardinality, pairing will leave one of the collections with objects that are unpaired and this can be used to define a size relationship between them. The collection in which all objects are paired is said to be "smaller" and the one left with unpaired objects "larger", than the other.'

This is what you edited in.

Ignore the first sentence since you cannot say if two collections do not have the same cardinality then they do not have (thus have greater) same cardinality.

Pairing CAN leave one of the collections with unpaired objects even if it's the same collection, pairing the natural numbers with the even natural numbers leaves the odd natural numbers unpaired.

You would have to say instead if it is impossible to pair without leaving unpaired, but that's saying if you cannot prove = that proves ≠ thus >. But lack of proof is not proof of it's opposite. If you believe that your wrong. And you wouldn't have a single definition for = and ≠ thus >. Simply if one can pair all of a set with only some of a set then the latter set is proven larger using the same sort of incorrect reasoning as your proof of equal size. But then for example the set of evens would be proven larger than the set of primes. So set theory DOESNT believe that simplicity.

Now why does Wikipedia very verbosely describe set theory contrary to Wikipedia policy, but not point out their reasoning flaw, assuming that which applies by proof to the finite, perfect one to one matching proving same size, applies to the same infinite without proof but rather presumption, one to one matching of infinite sets proves same size, but only SOMETIMES assuming that which apply to the finite by proof applies to the same infinite. that is for, as per the early literature, ≤ and ≥ and they say by proof only = but said proof obviously needs = defined and the ≤ and ≥ parts are also definitions (of cardinality), so correcting them, for ≤ = and ≥ but forbidding for < or > since then all sets in Aleph0 would be < and = and > each other in infinite size, likewise but separately all sets in Alreh1. That's not math itself ('crisis in mathematics') or the foundations of all math as Wikipedia claims but an individual manipulated system of math.

Likewise to prove > they take their proof of ≥ but need proof of ≠. Thus the diagonal argument. But it ONLY proves their proof of = won't work, for example between the naturals and the reals from 0 inclusive to 1 exclusive. So they assume lack of proof of = is proof of ≠.

But no the lack of countability is from the naturals are atomic and their version of the reals is nonatomic, being instead fluid. So an atomic set cannot be atomicly paired with a nonatomic set.

Likewise for ordinal 1+∞=same ∞ (omega) but ∞+1≠same ∞ so they say is ordinal omega +1. The first is adding to the beginning of by definition beginning but never ending list of (correcting them since allowing to continue without end would obviously continue to any defined ∞s) only finite numbers, the second adding to the end of said never-ending that is without end list. They assume since they cannot prove = size that means ≠ thus larger size, but no rather cannot add to non-existent end of list so cannot prove by matching a list to a nonlist.

Now

If Wikipedia is so verbose about set theory

Why not include all of it

Why not include the reasons for what is said in Wikipedia

Obviously because that would be embarrassing

Also

Why not explain things childishly simple

Obviously since then the errors would be figured out

But if you Wikipedia editors would rather just say 'these are the rules' then set theory is not math itself ('crisis in mathematics') not the foundations of math but a private game. Victor Kosko (talk) 02:06, 4 March 2026 (UTC)

Two collections have the same cardinality if it is possible to pair their constituent elements in some way; as you say, for infinite sets it is also possible to make a pairing from one set of a particular cardinality to a subset of another set of the same cardinality (e.g. to a subset of itself), which is famously counterintuitive for those used to discussing finite sets (cf. Hilbert's paradox of the Grand Hotel); some even reject the existence of "completed infinity", cf. Actual and potential infinity, but this is a philosophical rather than mathematical question.

In the context of a finite collection whose cardinality is a natural number, these counterintuitive ideas are not really relevant, putting further discussion mostly out of scope for this article. We could maybe set aside one or two sentences for a discussion of the cardinality of the whole set of natural numbers (or we could even add an entire section about it toward the end of the article), linking interested readers to Cardinality or Countable set for more information. –jacobolus (t) 03:59, 4 March 2026 (UTC)

The third paragraph in § Size of a collection, If two collections do not have the same cardinality, pairing will leave one of the collections with objects that are unpaired and this can be used to define a size relationship between them. The collection in which all objects are paired is said to be "smaller" and the one left with unpaired objects "larger", than the other., is only valid for finite sets; a minor edit should fix that. It might be reasonable to briefly mention infinite cardinals and ordinals in § Generalizations, with a nod to the Axiom of Choice. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:12, 4 March 2026 (UTC)

read the article. It goes into infinite set theory heavily to the extent that someone else deleted ALOT of material from the cardinality article.

The problem obviously it the editor is not qualified to edit set theory infinity material.

I want the set theory cardinality lies in one place and likewise ordinal, BUT as I said COMPLETE, that is with the embarrassing stuff. Your claiming the natural numbers article should leave out the set theory infinity stuff. Fine, except per the problems your having with the anti set theory people it should clarify in set theory there are no infinite natural numbers. Also another set theory error, you assume by defining the natural numbers the way you do there will automatically be no infinites in the set. Not so, it has to be an axiom, without any limit the sequence obviously will go to any defined infinity, there's nothing to stop. Victor Kosko (talk) 18:37, 4 March 2026 (UTC)

"read the article. It goes into infinite set theory heavily" – Which article are you talking about? The section you are complaining about says nothing beyond "Taken together the natural numbers themselves form an infinite sequence", which doesn't really seem controversial. –jacobolus (t) 19:17, 4 March 2026 (UTC)

I think the edits you made address the issue fully.

I used the ordinary language word "collection" rather than "set" to avoid unhelpful technicalities in this section.

IMO the ordinary meaning of "collection" is finite, but it doesn't hurt to make this explicit. TheGrifter80 (talk) 01:16, 6 March 2026 (UTC)

Position in a sequence misleading

The sentence A sequence is a collection of objects with a specific order, meaning that every object has a position and comes either before or after every other object according to a given criteria. in § Position in a sequence is misleading. $\mathbb {Q}$ and $\mathbb {R}$ match that description. I'm trying to come up with wording that is both accessible to the layman and correct, and the terms discrete and well ordered would seem to be too technical for an informal section. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:25, 3 March 2026 (UTC)

It's a fair point, but this section is giving an intuitive description of natural numbers, and I think your objection is beyond the scope here. Doesn't the word "object" in this context imply discreteness?

For example, if I have bag of oranges and want to order them by weight, I don't think this means you have consider the infinite ways you might cut them into pieces first. TheGrifter80 (talk) 02:51, 6 March 2026 (UTC)

Further to this: I fully agree that nothing in this section should be incorrect or misleading. The challenge is that naturals are logically prior to rationals and reals so it's difficult to give an understandable conceptual description in a way that simultaneously accounts for further extensions of the concept.

As I mentioned in a comment above I have used "collection of objects" which (IMO) means finite and discrete in ordinary language. But always keen to hear different views and suggestions for improvement. TheGrifter80 (talk) 11:14, 8 March 2026 (UTC)

I tried, the key was using a definition of sequence that did not assume the natural numbers (so one borrowed from ordinal numbers for transfinite sequences). The problem is it is a very technical workaround for what is a relatively intuitive observation. I'm not saying what I did is the final form, but I think it is an improvement, both in terms of Victor's complaints and in sourcing. Mathnerd314159 (talk) 22:27, 15 March 2026 (UTC)

I can see where you're going with that, but I agree it's too technical for an intuitive section. What do you think are the inaccuracies of the section as it previously was? If it is that some/all of the terms "sequence", "order" and "ordinal number" were used in a way that didn't strictly match their technical definitions, I think another approach would be to rephrase using only ordinary language, without implying any technical definitions. TheGrifter80 (talk) 00:58, 16 March 2026 (UTC)

The current version does not seem correctly targeted for the expected audience. I think we should instead start with a gloss along the lines of "A sequence is an ordered list of objects". (And we could even initially skip "ordered", which is perhaps included in the concept of a "list".) –jacobolus (t) 03:56, 16 March 2026 (UTC)

What about something like this to start with:

A sequence is an ordered list of objects. The natural numbers form an infinite sequence, meaning they have a fixed order, specific starting point and no end point: 1, 2, 3, and so on indefinitely.

Any finite collection of objects that is ordered in some way, such by size or weight, also forms a sequence that can be described using natural numbers. Each object in the sequence is identified with a number, meaning that it occupies the same position in the sequence of objects as that number occupies in the sequence of natural numbers.

TheGrifter80 (talk) 04:43, 16 March 2026 (UTC)

The typical use of natural numbers is actually for infinite sequences, e.g. you will see $a_{1},a_{2},\ldots$ which implicitly indexes an infinite sequence $a_{n},n\in \mathbb {N}$. Well, see what you think of the new version. Mathnerd314159 (talk) 16:44, 19 March 2026 (UTC)

Natural numbers are answers to questions like "how many..."

@Jacobolus: Regarding

Natural numbers are answers to questions like "how many apples are on the table?"

you wrote "this phrasing is confusing" when reverting. Could you elaborate on what is confusing here? If others can explain why this is confusing, that would be helpful too. Thank you, Ebony Jackson (talk) 01:24, 17 March 2026 (UTC)

An "answer" to the question would be a sentence such as "There are 4 apples on the table". Such a sentence is self evidently not a number. Saying that a number is an "answer" is going to confuse the hell out of a large number of readers. It confuses the hell out of me. Beyond that, even if you think most people will expext "4" to be the "answer to the question ...", you start running into nontrivial philosophical questions about the meaning/nature of numbers and other abstract objects that are well out of scope here, and I just don't think it's worth opening that can of worms. –jacobolus (t) 03:25, 17 March 2026 (UTC)

Let me turn it around, in case it helps to show why I find this construction potentially confusing. We might imagine a hypothetical conversation: A: "What is a natural number?" B: "It's the answer to a question about apples on a table." –jacobolus (t) 15:21, 17 March 2026 (UTC)

I could live with either version. but agree that saying numbers "are answers" feels a bit awkward. It seems much easier to say what numbers are used for than what they truly "are".

If the aim here is to use more direct language, perhaps "are used" instead of "can be used" is a small improvement.

Or, another option. Frege refers to:

positive whole numbers which give the answer to the question "How many?"

: TheGrifter80 (talk) 07:59, 17 March 2026 (UTC)

Can we replace the animations with static images?

@TheGrifter80 thanks for your nice images of apples and oranges. Those are clear and effective. I don't think the animation part is really that essential though, and may be distracting for some readers. Each of the two animations has one frame which I think effectively conveys the same message; can we just go for those as static images instead. (Aside: if you do want to keep an animation, try to make sure that the initial frame is as close to independent as a stand-alone static image as possible, since some readers may have animation disabled and articles can be presented in contexts where an animation is frozen, e.g. turned into a PDF file.) –jacobolus (t) 17:00, 20 March 2026 (UTC)

jacobolus, thanks - yes agree for the first image the animation doesn't add much so I'll revert to a static image. For the second one I was trying to convey the dynamic aspect of counting. Let me amend it so that the initial frame works as a standalone and let's see how that looks. If consensus is that it's better static, I can change. TheGrifter80 (talk) 23:39, 20 March 2026 (UTC)

For the "cardinality principle of counting" image, I too think that a static image would be better. Maybe instead of animation you could highlight the 3 in some way to indicate that it is the answer to the counting problem. Ebony Jackson (talk) 01:36, 21 March 2026 (UTC)

@jacobolus, @Ebony Jackson - How about this? If not, give me your suggestion or feel free to edit the file.

— Preceding unsigned comment added by TheGrifter80 (talk • contribs) 03:51, 21 March 2026 (UTC)

That looks pretty good, though I might scoot things to the left and not leave the "4..." quite so pushed into the margin. Also, is it just me or is the middle arrow not quite centered between the other two? –jacobolus (t) 04:15, 21 March 2026 (UTC)

I too think it looks pretty good. My own personal nitpick would be to ask that 1,2,3,4 be equally spaced instead of having the distance between 3 and 4 be greater! Ebony Jackson (talk) 04:33, 21 March 2026 (UTC)

I'll thought it was even, but I'll double check. Does it need the "4..." or would it be better with "3..."? TheGrifter80 (talk) 05:42, 21 March 2026 (UTC)

I think it is better with the 4..., or maybe even 4 5... It makes it clearer that on the bottom we have ALL the natural numbers. Ebony Jackson (talk) 15:51, 22 March 2026 (UTC)