Talk:Natural number

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Article seems to confuse two different concepts

I don't think it makes sense to discuss both the non-negative integers and the positive integers in the same article, when integers gets its own article. The difference may only be a single element (namely, zero)—but this single member is of great importance, e.g. when discussing factorization, being that, while it's the additive identity, it's also the multiplicative annihilator. You do not want to define a domain of a function and get these two mixed up.

Further, this article and Whole number both suggest that "whole number" and "natural number" are synonyms, but there is plenty of literature that use both terms, in these cases, they have distinct definitions, usually "natural number" excluding zero.

Given this fact, Wikipedia is not a dictionary suggests that when multiple distinct concepts can be isolated, they should get their own article. The only exception being if the article "discusses the etymology, translations, usage, inflections, multiple distinct meanings, synonyms, antonyms, homophones, spelling, pronunciation, and so forth of a word or an idiomatic phrase" which is clearly not the case here. The fact that the layman sometimes uses terms interchangeably, instead of in the meaning isolated by the articles, is not an excuse.

This article is not a discussion of the word, or the term, it is a discussion of a specific mathematical concept. The fact that, historically, the non-negative integers (or integers, or positive integers) has at times been called the "whole numbers" and/or "natural numbers" is a fact to list in the relevant article whatever its name may end up being; this does not make an excuse to combine the two concepts into the same article.

I see two obvious corrections, either all three concepts should share Integer, which would discuss related subsets and how that affects their mathematical properties; or the relevant portions of this article moved to Whole number. Alternatively, appropriate sources could be added to show why these two sets are, in fact, the same mathematical concept deserving of a single article. Awwright (talk) 05:37, 17 June 2024 (UTC)[reply]

<tired sigh>

Look, of course the two sets are not literally the same thing. That's not the point. The point is that there's very little that we want to say about the natural numbers that depends on whether or not zero is included.

As for "whole number", that's a term that is not much in use in research mathematics. --Trovatore (talk) 05:57, 17 June 2024 (UTC)[reply]

I mentioned it up above, there is actually quite a lot that can be said. What you call "very little" is more than most articles, e.g. I just clicked random page and got Usina do Gasômetro, it is 3 paragraphs. There are definitely 3 paragraphs of information each about the positive integers and the non-negative integers. And there are definitely a ton of reliable sources, so they are both notable. To me it is obvious they should be separate articles. But apparently 8 people disagree. 8 vs. 2 now, maybe it is time for another split proposal. 😄 Mathnerd314159 (talk) 06:30, 17 June 2024 (UTC)[reply]

"Can be said" is not the same as "want to say". Sure, there's a fair amount you could potentially say about off-by-one errors, but it doesn't live naturally in an article about the natural numbers.

Basically no one studies "the natural numbers with zero" and "the natural numbers without zero" as distinct objects of study. Sure, occasionally you will find someone who has symbols for both of them, but that is not the same thing. The natural numbers are an incredibly rich mathematical structure, the study of which has been the principal preoccupation of the entire professional lifetimes of many many brilliant people. None of those people ^[a] divide that study into the structure with or without zero. They pick one for definiteness, but recognize that everything they say would translate with minor changes to the other convention. --Trovatore (talk) 01:33, 18 June 2024 (UTC)[reply]

The situation here is kind of similar to what you describe with the off-by-one, most of the article is about nonnegative integers and then there is some stuff about positive integers unnaturally mixed in. It is true no one studies "the natural numbers with zero" and "the natural numbers without zero", but that is because they are unnatural terms. There are plenty of textbooks that define positive integers and nonnegative integers as distinct objects of study and use them precisely.

I don't agree that the natural numbers are a mathematical structure. A mathematical structure has one definition but the natural numbers have two - no set both contains and does not contain 0. And I would argue that each paper's picking a definition does divide the literature up. As soon as you get past the basic Peano axioms, nothing translates without major changes or adding ugly conditions like "≠0" - for example, exponentiation on positive integers is well-defined, but 0^0 is not. If it really was completely equivalent there would not be a debate, there would be a theorem. Mathnerd314159 (talk) 04:40, 18 June 2024 (UTC)[reply]

OK, you're again descending into quibbles that make it hard for me to believe you're taking this seriously. --Trovatore (talk) 05:18, 18 June 2024 (UTC)[reply]

Mathnerd, you're wasting your time here. You've repeated the same couple of points now ad nauseam, while throwing in a mishmash of irrelevant apples-to-oranges comparisons, non sequiturs, and straw men, but it's not convincing anyone. If 8 people trying to explain why this seems like a bad idea was too few for you to get the point, you are welcome to canvass WT:WPM where you can probably get another 10 or so Wikipedians to voice their disagreement with you. Or you can take your discussion to twitter or something. It's not going to accomplish anything though. –jacobolus (t) 07:06, 18 June 2024 (UTC)[reply]

The fact that a few things depend upon one's choice of convention doesn't mean that there are two separate concepts or that splitting the explanation across two pages would help anyone learn. XOR'easter (talk) 16:56, 18 June 2024 (UTC)[reply]

1. This sort of categorical statement is always risky; I imagine you can find someone who has both done good work and also claims to make an important distinction, but such a person would at the very least be an outlier

Phrasing dispute

So regarding the edits by User:121.211.95.94... they actually seem reasonable? Specifically the edits are:

−

Sometimes, the '''whole numbers''' are the natural numbers as well as zero.

+

Sometimes, the '''whole numbers''' are the positive integers as well as zero (that is, the non-negative integers).

−

For example, the [[integer]]s are made by adding 0 and negative numbers.

+

For example, the [[integer]]s are made by adding 0 (if not already included) and negative numbers.

These both seem in keeping with the concept that 0 may or may not be considered a natural number. I've been reading the article several times like @Remsense suggested and they still look like good edits. Well, the whole numbers could be shortened to "Sometimes, the whole numbers are the non-negative integers." as the non-negative integers are already defined. Mathnerd314159 (talk) 05:51, 29 April 2025 (UTC)[reply]

I suspect the point here is that sources that use the phrase "whole numbers" in this meaning generally do not include zero in the natural numbers, since then there would be no reason to use both terms. If that is the intent, I have to say I think phrasing could be found that makes the point more directly and concisely; I also struggled to figure out what people's objection was to 121's changes. Maybe something like some texts use the term whole numbers to refer to the set with zero included, and natural numbers to refer to the set without zero. Just a first cut; I don't completely love it but I think it's better than what's there now.

As an aside, I would prefer not to say "integers" too often in the lead section. I'm not worried about logical circularity, but I think it might be hard to follow for some readers — they come here to learn about natural numbers, and they get shunted off into a discussion of integers, and it might be extra stuff to keep track of. --Trovatore (talk) 06:50, 29 April 2025 (UTC)[reply]

The problem with IP's formulation is that it uses the integers just before defining them. D.Lazard (talk) 07:35, 29 April 2025 (UTC)[reply]

Aye, this is what I saw. It's a shame we never got to that point because someone decided it was investigation o'clock. Remsense ‥ 论 07:44, 29 April 2025 (UTC)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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., ${\displaystyle \omega +1=\omega \cup \{\omega \}}$,^[a] the second infinite ordinal. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 15:10, 24 September 2025 (UTC)[reply]

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)[reply]

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

Notes

1. "${\displaystyle \omega }$" is the conventional notation for the first infinite ordinal, the order type of ${\displaystyle \mathbb {N} }$.

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

@Stepwise Continuous Dysfunction: well there is [1]: "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)[reply]

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)[reply]

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

The set of the partitions of a natural number ⁠${\displaystyle n}$⁠ and the symmetric group ⁠${\displaystyle S_{n}}$⁠ 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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

How is In contrast, number theory,^[1] analysis,^[2] dictionaries,^[3][4] and most schoolbooks (through high-school level)^[5] 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^[6] 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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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^[6][7] in the same field(s). -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:48, 16 October 2025 (UTC)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

References

1. Křížek, Michal; Somer, Lawrence; Šolcová, Alena (21 September 2021). From Great Discoveries in Number Theory to Applications. Springer Nature. p. 6. ISBN 978-3-030-83899-7.
2. See, for example, Carothers (2000, p. 3) harvtxt error: no target: CITEREFCarothers2000 (help) or Thomson, Bruckner & Bruckner (2008, p. 2) harvtxt error: no target: CITEREFThomsonBrucknerBruckner2008 (help)
3. "natural number". Merriam-Webster.com. Merriam-Webster. Archived from the original on 13 December 2019. Retrieved 4 October 2014.
4. Cite error: The named reference MacTutor was invoked but never defined (see the help page).
5. Cite error: The named reference Enderton was invoked but never defined (see the help page).
6. Keisler, H. Jerome (June 19, 2022). Foundations of Infinitesimal Calculus (PDF).
7. One of Carothers (2000, p. 3) harvtxt error: no target: CITEREFCarothers2000 (help) or Thomson, Bruckner & Bruckner (2008, p. 2) harvtxt error: no target: CITEREFThomsonBrucknerBruckner2008 (help)

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)[reply]

@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)[reply]

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)[reply]

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)[reply]

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)[reply]

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

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

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]

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)[reply]