Talk:Natural number

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,^[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)

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^[6][7] 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)

@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)

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 ${\displaystyle \mathbb {Z} }$, ${\displaystyle \mathbb {Q} }$, ${\displaystyle \mathbb {R} }$, ${\displaystyle \mathbb {C} }$, ${\displaystyle \mathbb {H} }$, ${\displaystyle \mathbb {O} }$. 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)

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)

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 ${\displaystyle (\mathbb {N} ,+)}$ 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 ${\displaystyle \times }$ can be defined via a × 0 = 0 and a × S(b) = (a × b) + a. This turns ${\displaystyle (\mathbb {N} ^{*},\times )}$ 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)

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., ${\displaystyle \omega }$ and ${\displaystyle \omega +1}$ 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)

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. ${\displaystyle \mathbb {Q} }$ and ${\displaystyle \mathbb {R} }$ 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 ${\displaystyle a_{1},a_{2},\ldots }$ which implicitly indexes an infinite sequence ${\displaystyle a_{n},n\in \mathbb {N} }$. Well, see what you think of the new version. Mathnerd314159 (talk) 16:44, 19 March 2026 (UTC)

@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)

@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)