Talk:Natural number
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Formal and intuitive définitions.
[edit]In the introduction, the present definition is not a real mathematic one, but rater an intuitive one. In line with the french version, I propose to add, in front of the present text, formal definitions in the frame of axiomatic arithmetic and in the frame of set theory: « In mathematics natural number means formally:  a primitive notion of axiomatic arithmetic.  a settheoric construction satisfying Peano axioms.
Intuitively, the natural Numbers … » CBerlioz (talk) 11:18, 9 September 2021 (UTC)
 The mathematical definition of natural numbers does not belong to the first sentence. Please read WP:TECHNICAL for a detailed explanation of this assertion. The concept of natural numbers predates its formalization for many centuries, and, presently, billions of people use natural numbers without knowing their formalization, and even without knowing what a formalization is.
 Also, your formulation is wrong: "means formally" is an oxymoron, as "means" refers to an explanation (that is not a proof), while "formally" refers to proofs that are generally the opposite of an explanation. Moreover boths items of your definition are wrong: 1/ There is no common axiomatization of arithmetic for which each natural number is a primitive notion. 2/ Set theory construction and Peano axioms provide two different formal definitions of the same concept of a natural number, that have be shown to be equivalent. 3/ Itemization suggests wrongly the existence of two different concepts. D.Lazard (talk) 11:54, 9 September 2021 (UTC)
 No, as the sentence refers to the most technical content of the article, it should be at the end of the lead. D.Lazard (talk) 11:39, 11 September 2021 (UTC)
You are right when you stress that billions of people use natural numbers without knowing their formalization. Counting is a basic human skill. Kronecker wrote : God made the integers and all the rest is the work of man. But this article is not under the portal of Psychology, it is under the portal of Mathematics and it actually deals with formalization, notably under the subtitle 4 « Formal definitions », This should be announced, if not in the first sentence, elsewhere in the introduction.
In an axiomatic theory of arithmetic, the notion of natural number (and not each natural number) is by definition a primitive notion, otherwise it would be an axiomatic theory of sets or an axiomatic theory of any other objectif. The natural numbers constructed within set theory must be distinguished from those of axiomatic arithmetic because they are terms of different theories, as informal natural numbers must be distinguished, even they have (happily) great similarities.
My new proposal is: « In informal mathematics, the natural numbers are those … in a mathematical sense.
In formalized mathematics, the natural numbers are both:  the terms of axiomatic arithmetic.  a construction of set theory satisfying the Peano axioms.
The set of … » CBerlioz (talk) 10:43, 10 September 2021 (UTC)
 The distinction between formal and informal mathematics is your own invention (formal and informal reasoning both belong to mathematics, and a large part of mathematicians activity is to infer informally some results and then to prove them formally). So this distinction has not its place here, and I strongly oppose your suggestion. D.Lazard (talk) 11:00, 10 September 2021 (UTC)
 I agree with D.Lazard. This article is about the natural numbers, which existed long before mathematicians began trying to formalize them. And while such formalizations are an important aspect of the topic of this article, the article should not lead with them. Paul August ☎ 11:50, 10 September 2021 (UTC)
I don’t want to start a dispute between formalists and intuitionists. May we agree on the addition in the introduction of the more neutral sentence: « The notion of natural number has been formalized on one hand by axiomatic arithmetic of which natural numbers are the terms, and on the other hand in set theory by constructing terms called finite ordinals. »? CBerlioz (talk) 09:24, 11 September 2021 (UTC)
 This is definitively not for the beginning of the lead. However, one could add the following at the end of the lead:
The definition of natural numbers has been formalized in several essentially equivalent ways, through Peano's axioms or set theory
. This is less technical than your formulation, and reflects better the content of the relevant section and subsections. The fact that these formalizations are essentially equivalent deserves to be explained precisely at the beginning of the section § Formal definitions. D.Lazard (talk) 09:53, 11 September 2021 (UTC)
I agree with your simpler formulation. I think it should logically take place just before « The natural numbers are a basis from which many other number sets … ». CBerlioz (talk) 11:23, 11 September 2021 (UTC)
 No, as this refers to the most technical part of the article, it must be at the end of the lead. For the same reason the corresponding section is at the end of the article. D.Lazard (talk) 11:39, 11 September 2021 (UTC)
OK, but the paragraph « The natural numbers are a basis …in the other number systems. » should also be at the end of the lead for the same reason, and perhaps developped in a new section. CBerlioz (talk) 13:39, 11 September 2021 (UTC)
Finally, I think it would be better and less technical to refer to modern definitions ( second item of the section History) rather than to formal definitions: « Modern definitions of natural numbers are based on several essentially equivalent approaches, through set theory or Peano’s axioms. » CBerlioz (talk) 08:52, 14 September 2021 (UTC)
 I am strongly against this formulation in the lead: it suggest wrongly that modern definitions differs from older ones; in fact, Peano's approach is simply a formalization of the old concept of ordinal numbers, and the set theoretic approach is a formalization of the older concept of cardinal numbers. It is because the concept of formalization is unknown to many people that its need must be explained. In summary, my opinion is that the current lead is the best that we can get without input of other editors, and that it must be left unchanged without such inputs. D.Lazard (talk) 15:51, 19 September 2021 (UTC)
 I agree with D.Lazard. As I've said above, there is no need to mention formal definitions in the lede. Paul August ☎ 20:20, 19 September 2021 (UTC)
Is it possible to paraphrase "Intuitively" as other words ? Because it seems to mean Intuitionism or Intuitionistic logic.SilverMatsu (talk) 06:10, 20 September 2021 (UTC)
 I don't see why anyone would confuse intuitively with intuitionism or intuitionistic logic. I think most readers will understand the former and may not have even heard of the latter.—Anita5192 (talk) 19:58, 20 September 2021 (UTC)
Following deletion by Trovatore of the latest contribution of D.Lazard I suggest: « Modern definitions of natural numbers formalize the older intuitive ones of cardinal or ordinal through set theory or Peano axioms (of which natural numbers are a primitive notion). » CBerlioz (talk) 16:39, 24 September 2021 (UTC)
 I agree with Trovatore's deletion and his assertion that this kind of sentence is not appropriate for the lead. D.Lazard (talk) 17:24, 24 September 2021 (UTC)
 And me as well. Paul August ☎ 20:56, 24 September 2021 (UTC)
Alternative places are the head of Modern definitions section or the head of Formal definitions section. CBerlioz (talk) 10:58, 25 September 2021 (UTC)
 Before discussing where placing a sentence, a consensus is needed for establising whether such a sentence improves the article. IMO, this is not the case, as this is already discussed in details in § Modern definitions. Please, stop triyng to push your point of view against a clear consensus (three editors against you). D.Lazard (talk) 11:53, 25 September 2021 (UTC)
Existence of the set of natural numbers
[edit]For speaking about the set of natural numbers, its existence must be admitted. This is the role of the different forms of the axiom of infinity. The simplest one is: there exists a set that contains all natural numbers. It is adapted to the case where natural numbers are introducted before set theory (for exemple Fraenkel in Abstact Sets). CBerlioz (talk) 13:23, 23 September 2021 (UTC)
 "For speaking about the set of natural numbers", the concept of a "set" must be defined, but one can define and use natural numbers without talking of the set of natural numbers (this has been done by mathematicians during more than 2,years). This article is about natural numbers, not about the set formed by them. So, please, do not add subtilities that cannot be completely clarified without referring to the foundations of mathematics and the logical technicalities that they involve. D.Lazard (talk) 13:46, 23 September 2021 (UTC)
The problem is that the article talks of the set of natural numbers. Do you suggest to delete any mention to it? If not, the reader must know or learn that some sets don’t exist because their existence would lead to contradictions. I tried to make my sentence as less technical as possible. CBerlioz (talk) 15:30, 23 September 2021 (UTC)
 The article uses the naive concept of a set, and the logical questions of consistency of set theory do not matter in it. So, care is needed for writing the article for being correct at the elementary level as well as at the advanced level of specialists of set theory.
 Moreover, the sentence that I have reverted is contradictory with the article:
Peano arithmetic is equiconsistent with [...] ZFC with the axiom of infinity replaced by its negation.
This means that the assertion "the natural numbers form a set" is not a consequence of Peano's axioms: if they would form a set, there would exist an infinite set.  In summary, although there are infinitely many natural numbers, the axiom of infinity is not required for defining natural numbers. In other words, Peano's axioms do not allow to talk of "the natural numbers" as a whole. D.Lazard (talk) 16:39, 23 September 2021 (UTC)
We are in agreement on the fact that the axiom of infinity is independant of Peano axioms. That is the reason why it must be added when set theory is considered as an extension of arithmetic. Perhaps would it be simpler to add the sentence at the end of the paragraph settheorical definitions, instead of the end of the section modern definitions: the existence of the set of natural numbers is guaranteed by the axiom of infinity ? CBerlioz (talk) 17:27, 23 September 2021 (UTC)
 No, these considerations do not belong to such an elementary article. Moreover you seem to not have a source for the assertion that you want to add (see WP:OR). D.Lazard (talk) 17:55, 23 September 2021 (UTC)
 It's also problematic to speak of "defining" the natural numbers via axioms. Axioms do not define; they axiomatize. It's true that there is (up to isomorphism) only one model of the Peano axioms using full secondorderlogic semantics, and that could be taken as a definition, but while this might fit in the body somewhere, it's not appropriate for the lead. Trovatore (talk) 18:04, 23 September 2021 (UTC)
It’s quite elementary to assess or prove the existence of a set (for example Paul Halmos, Naïve Set Theory). CBerlioz (talk) 08:51, 24 September 2021 (UTC)
 What is the above comment responding to? Trovatore (talk) 21:08, 24 September 2021 (UTC)
As long as the existence of the set of all natural numbers has not been assessed or proved, you must speak of the class of all natural numbers instead of the set of all natural numbers. CBerlioz (talk) 08:22, 7 September 2022 (UTC)
 In this section, readers are not supposed to know that there are differences between "collection", "class" and "set", and there is nothing wrong in the present formulation for people who know the difference Moreover, the notation (that is the subject of the section) is independent from the fact that the natural numbers form a set or not. It is pedantry to introduce here advanced concepts of set theory.
 Also, your formulation is logically wrong. It is not the existence of the set of natural numbers that could require a proof, it is the property that they form a set (the existence of the natural numbers is the subject of the whole article). So, your formulation ("as for the existence of such a set, see ...") should be read as "for the proof that the natural numbers form a set, see ...). So, your link is wrong, and, again, this does not belong to this section. D.Lazard (talk) 10:30, 7 September 2022 (UTC)
Would you agree with the formulation: “Naïve set theory admits that the natural numbers form a set, to which mathematicians refer …” ? CBerlioz (talk) 10:54, 8 September 2022 (UTC)
 Definitively not. This is not useful here and may be confusing for many readers. The present formulation is mathematically correct, and needs not to be changed. Moreover the fact that natural numbers form a set is more or less already explained in the preceding sections. D.Lazard (talk) 11:25, 8 September 2022 (UTC)
The fact that natural numbers form a set is rather less than more explained in the preceding sections. It should be explicitly stated, for example by the following formulation in Modern definitions section, quoting the definition of N.Bourbaki or P.Suppes: “… a particular set, named cardinal, and any set … to have that cardinal. The set of natural numbers is then defined as the set of finite cardinals.” Have you a better solution for stating that natural numbers form a set ? CBerlioz (talk) 08:18, 9 September 2022 (UTC)
Another (non exclusive) solution could be at the beginning of 3rd paragraph:
“Natural numbers form a set. Many other number sets are built by successively extending the set of natural numbers: ….” CBerlioz (talk) 11:40, 12 September 2022 (UTC)
 Done, with the article "the" added and number set linked D.Lazard (talk) 13:33, 12 September 2022 (UTC)
Construction based on cardinals could be added in section Formal definitions. CBerlioz (talk) 07:31, 13 September 2022 (UTC)
"N (math)" listed at Redirects for discussion
[edit]An editor has identified a potential problem with the redirect N (math) and has thus listed it for discussion. This discussion will occur at Wikipedia:Redirects for discussion/Log/2022 July 28#N (math) until a consensus is reached, and readers of this page are welcome to contribute to the discussion. –LaundryPizza03 (dc̄) 03:57, 28 July 2022 (UTC)
Constuctions based on set theory
[edit]The construction based on cardinals could be added, for example with the following wording:
“The simpler way to introduce cardinals is to add a primitive notion, Card(), and an axiom of cardinality to ZF set theory (without axiom of choice).
Axiom of cardinality (P.Suppes): The sets A and B are equipollent if and only if Card(A) = Card(B)
The definition of a finite set is given independently of natural numbers:
Definition (Tarski): A set is finite if and only if any non empty family of its subsets has a minimal element for the inclusion order.
Theorem: If a set A is finite, any set equipollent to A is finite.
Definition: a cardinal n is a natural number if and only if there exists a finite set x such that n = Card(x)” CBerlioz (talk) 16:36, 19 September 2022 (UTC)
 This article has already a section § Constructions based on set theory with a link to Settheoretic definition of natural numbers. If you want to add a new definition, it must be reliably sourced from a textbook, and you must provide some evidence that this definition is often considered. It seems that this is not the case of your approach. D.Lazard (talk) 17:13, 19 September 2022 (UTC)
This approach is used in Patrick Suppes, 1972 (1960), Axiomatic Set Theory. Dover. Natural numbers are also defined as finite cardinals in N.Bourbaki, 2006 (1970) Elements de Mathématique Théorie des ensembles, Springer Berlin Heidelberg New York. Axiomatic definition of cardinals is also used in A.Fraenkel 1968 (1953) Abstract set theory, NorthHolland Amsterdam. CBerlioz (talk) 11:04, 20 September 2022 (UTC)
 The § von Neumann definition given in this article is based on cardinals as well as on ordinals, since only finite sets are considered here, and the number n is defined as a set of n elements. It is because this section did not comply with the manual of style that it seemed to be based on ordinal theory. So, I have edited it for being clearer for nonspecialists, and removing the emphasis on ordinals.
 Your definiton "a cardinal n is a natural number if and only if there exists a finite set x such that n = Card(x)" is much too technical for this article: For finite sets, "the cardinal of a set is n" is a pedantic way to say "the set has n elements". So, all your advanced considerations, could be replaced in this article by "with von Neumann's definition of the natural numbers, a set S has n elements if there is a bijection from n to S". D.Lazard (talk) 15:17, 20 September 2022 (UTC)
 I have added this to the article. D.Lazard (talk) 15:29, 20 September 2022 (UTC)
I agree with this simplification. A further simplification could be the replacement of “Constructions based …” by “Construction based ...’’ with the deletion of Zermelo ‘s definition, which has only a historical interest. CBerlioz (talk) 07:46, 21 September 2022 (UTC)
 It is difficult to temove the mention of Zermelo‘s definition, since it is the trget of a redirect. So, I have merged the two subsections of “Construction based ...’’ into a single section § Settheoretic definition. I have also added an introduction to § Formal definitions for explaining the relationship between the two approaches. D.Lazard (talk) 11:45, 21 September 2022 (UTC)
Problem with starting from 1
[edit]Considering the natural numbers to be cardinals and ordinals only makes sense if includes zero, since the null set has cardinality 0 and order type 0. Should the article discuss that or is that TMI? Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:21, 21 March 2023 (UTC)
 This is already discussed at the end of § Modern definitions. D.Lazard (talk) 12:29, 21 March 2023 (UTC)
 No, it is not. In fact, the article doesn't have the term null set at all. Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 21 March 2023 (UTC)Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 21 March 2023 (UTC)
 Null set is a concept of measure theory, and it is normal that it is not linked to in this article. The set of cardinality zero is called the empty set. It is linked to in the article, and used at least four times. D.Lazard (talk) 15:39, 21 March 2023 (UTC)
 Side note here that may be useful background information for current or future discussants. In American primary and secondaryschool textbooks, it is common to refer to the empty set as "the null set". Not sure how this got started, but it's not an inherently implausible name; it's just not the terminology used by mathematicians. American textbooks are also probably the reason that we get editors insisting that the natural numbers do not include zero, but the "whole numbers" do.
 Side note to the side note: In case anyone is wondering why sets are appearing in primaryschool textbooks, that's a legacy of the socalled New Math, a wellintentioned project to teach mathematics in a more conceptual and rigorous way from the start, which ran into the twin problems that the abstraction may have gone beyond what the children were developmentally ready for, and there may not have been enough teachers who understood it well enough to teach it effectively. Trovatore (talk) 16:58, 21 March 2023 (UTC)
 I'm not familiar with contemorary texts, but the ones I'm familiar with, e.g., Rudin,^{[1]} do not use the term null set for a set of measure 0 and various online sources, e.g., Britanica,^{[2]} MathWorld,^{[3]}^{[4]} list null set as synonymous with empty.
 Null set is a concept of measure theory, and it is normal that it is not linked to in this article. The set of cardinality zero is called the empty set. It is linked to in the article, and used at least four times. D.Lazard (talk) 15:39, 21 March 2023 (UTC)
 No, it is not. In fact, the article doesn't have the term null set at all. Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 21 March 2023 (UTC)Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:33, 21 March 2023 (UTC)
 That said, would you accept a footnote in the lead that cardinal and ordinal have more general meanings? Shmuel (Seymour J.) Metz Username:Chatul (talk) 20:05, 22 March 2023 (UTC)
 Zero is not a natural number, as generally in standard education in all countries (at least that I know of, e.g. Japan and Australia) consider natural numbers to begin from 1. Ztimes3 (talk) 07:11, 17 March 2024 (UTC)
 I believe that the relevant sources are University level Mathematics texts and peer reviewed Mathematics papers. K12 texts are typically not written by SMEs.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:41, 18 March 2024 (UTC)
References
 ^ Rudin, Walter (1976). Principles of Mathermatical Analysis. International Series in Pure and Applied Mathematics (Third ed.). McGraw Hill. ISBN 007054235X.
 ^ "Formal Logic  Set Theory". Britanica.
A class with no members, such as the class of atheistic popes, is said to be null. Since the membership of all such classes is the same, there is only one null class, which is therefore usually called the null class (or sometimes the empty class); it is symbolized by Λ or ø.
 ^ "Empty Set". MathWorld. Wolfram Research.
 ^ "Null Set". MathWorld. Wolfram Research.
Finite
[edit]I added the word finite and D.Lazard reverted it with the comment Too technical for the firsst sentence: this article is not primarily for those who knows infinite numbers. People who know infinite numbers already know natural numbers
. I believe that without finite in the first sentence the second sentence, Numbers used for counting are called cardinal numbers, and numbers used for ordering are called ordinal numbers.
, is misleading and should be removed.
Note that cardinals and ordinals are discussed later, in #Generalizations. Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:44, 21 March 2023 (UTC)
 The quoted sentence does not say that all cardinal numbers and ordinal numbers are finite. So the sentence is correct and not misleading for every interpretation of "cardinal" and "ordinal" (being a cardinal and an ordinal is a property of natural numbers)
 Nevertheless the formulation suggests that there are two sorts of natural numbers. So, I have changed the formulation of the sentence to clarify this point. D.Lazard (talk) 16:04, 21 March 2023 (UTC)
 So I think the purpose of mentioning the cardinal numbers/ordinal numbers in the lead is to explain that the natural numbers are sometimes referred to as the cardinal / ordinal numbers  the terms are used in a loose sense that doesn't allow infinities like the precise mathematical definition. If this is the case then it probably shouldn't link to cardinal/ordinal number as those are not the intended meanings. Mathnerd314159 (talk) 17:58, 21 March 2023 (UTC)
Is 0 a natural number?
[edit]I think it's common ground that we should show 0, 1, 2... as one meaning of "natural number". Is this the only definition, or should we also show 1, 2, 3...? If the latter, do we make one of them the primary meaning or give equal prominence? Certes (talk) 20:38, 18 February 2024 (UTC)
 Well, if we do assign a primary meaning, then I would say that meaning would be the one that included 0, given that it's ISO standard and used by most mathematicians. It's tricky though, even limiting to recentlypublished sources there are ones that say it is a matter of definition and that 0 is not a natural number. So the question is whether sources like these are sufficiently in the minority that discussing the "old" 1based definition in a section gives it sufficient weight, or whether it needs to be in the lead. Another solution would be to make a separate page for "counting numbers" or "positive integers" and have a hatnote  there are sufficient sources for notability. We could even do something like two pages "Nonnegative integers" and "Positive integers" and have "Natural numbers" be a DAB or a WP:Broad concept. Mathnerd314159 (talk) 00:24, 19 February 2024 (UTC)
 Previous discussions:
 Talk:Natural_number/Archive_1#Zero
 Talk:Natural_number/Archive_1#Lack_of_references (tail end)
 Talk:Natural_number/Archive_2#About_zero_as_natural_number
 Talk:Natural_number/Archive_2#Do_number_theorists_start_the_natural_numbers_with_1?
 Talk:Natural_number/Archive_2#Conventions
 Talk:Natural_number/Archive_2#Zero
 Talk:Natural_number/Archive_2#Modern_Convention
 Talk:Natural_number/Archive_3#Quotes_from_prealgebra_books
 Talk:Natural_number/Archive_3#Problems_with_this_article
 Talk:Natural_number/Archive_4#Where_do_we_start?
 Talk:Natural_number/Archive_4#Distinction_between_whole_numbers_and_natural_numbers
 Talk:Natural_number/Archive_4#Edit_request
 Talk:Natural_number/Archive_4#Problem_with_notations_:_the_notations_of_this_page_do_not_respect_international_standard_iso_notation
 Another idea is that, since I would say the current wording favors the 1based definition, to bias it towards 0, for example "the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0." Mathnerd314159 (talk) 01:01, 19 February 2024 (UTC)
 Ok 2A02:8070:6284:5100:F54B:87CF:FAC4:5CFB (talk) 13:08, 24 May 2024 (UTC)
 I would not consider 0 is not a natural number to be a RS and the nomenclature, e.g., "counting", "whole", is not that of Mathematics, but rather that of elementary public education. Further, the source is not consistent; one paragraph says that 0 is not a natural and the next says that it depends on the definition.
The natural numbers are so much a part of modern mathematics that they have their own special symbol, called a blackboard N. Similar symbols are available for the Integers, the Rational number, The Real numbers, and the Complex numbers.
Actually, the "old" definition did include 0. Zero was dropped for several reasons. First, most induction starts with 1, not 0, so you can begin an induction proof with "Let n be a natural number...". Second, if you drop zero, you can define a common fraction as having an integer numerator and a natural number denominator, and not have to specifically exclude 0. There are other reasons, but it is arbitrary, just as the order of operations is arbitrary, and some people swear by one version and some by the other.
In my experience, most books begin the natural numbers with 1. Rick Norwood (talk) 11:35, 19 February 2024 (UTC)
 If natural numbers are used as ordinal numbers (that is, for counting and enumerating), it is natural to start them from 1, since nobody (but mathematicians) would naturally talk of the 0th element of a sequence (what is the zeroth person entering in a room?). On the opposite, if the natural numbers are used as cardinal numbers, it is natural to start them from zero (there are zero person in the room). IMO the fact that both choices are commonly used in elementary textbooks results from the main focuses on natural numbers that are considered. In more advanced mathematics, starting from zero seems more common (natural?), since this simplifies many notations (for example, the zeroth term of a power series is the one that contains the zeroth power of the variable).
 So, I support Mathnerd314159's suggestion: "the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0." D.Lazard (talk) 15:40, 19 February 2024 (UTC)
 @Rick Norwood I don't see any evidence that an "old" definition included 0 (other than really old definitions that use "natural number" in an informal sense to mean all the integers). For example [1] [2] [3] [4] [5] all use 1, these span from 1800's to 1966. There are several references that attribute the 0based numbering to Bourbaki 1968 [6] [7].
 As for the symbol, this says it was Dedekind. And then Peano adopted N={1,2,...} from Dedekind, but later he used N+. So purely based on history, {1,2,...} should be denoted N+, leaving room for N={0,1,...}.
 As far as your "justifications" for "dropping zero", they sound like WP:OR. There is no common sense on Wikipedia so to argue that 1 is the "correct" definition you would need a reliable source that compares the two conventions and says starting at 1 is better. Now there are several sources in the article which justify starting at 0, e.g. the aforementioned ISO standard. There is some justification for 1 in the matter of definition source but they only end up with a weak preference for starting at 1. IMO what is really needed for the 1based definition to stay in the lead is a strong argument for starting at 1 so that we can assert that neither convention is in the majority of reputable sources.
 The question of whether "most" books start at 1 is certainly relevant, but again experience is a poor source of evidence. I had to spend some effort to find those 1supporting sources I gave. Looking at recentlypublished books, it seems split between universitylevel (which uniformly uses 0) and elementary school (which defines natural numbers as equivalent to "counting numbers" 1,2,3 and uses "whole numbers" for 0,1,2,3). I would classify the elementarylevel sources as "trash" and say that the universitylevel 0based convention constitutes the majority of reliable sources, but maybe others disagree. Mathnerd314159 (talk) 17:28, 19 February 2024 (UTC)
 So to illustrate what I mean by "removing 1 from the lead", it would be something like this: "In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., also called the nonnegative integers. Outside of mainstream mathematics, there exists a differing interpretation among some individuals, who define the natural numbers to encompass the positive integers (1, 2, 3, etc.)." Mathnerd314159 (talk) 17:43, 19 February 2024 (UTC)
 To quote page 17 of the Princeton Companion to Mathematics of 2008, "The natural numbers, otherwise known as the positive integers are ... 1, 2, 3, 4, and so on. ... (Some mathematicians prefer to include 0 as a natural number as well: for instance, this is the usual convention in logic and set theory. Both conventions are to be found in this book, but it should always be clear which one is being used.)" I guess that the editors were unable to get the contributors to the companion to agree to use one convention or the other. JonH (talk) 19:43, 19 February 2024 (UTC)
 I'd say that source is less useful than the "matter of definition" source  it doesn't give any arguments for using 0 or 1, merely gives the 1based first and then contradicts itself. And as it's from 2008, it's rather old  the age raises the question of whether there has been a shift towards 0based or 1based more recently. In my searches I deliberately set the filter to 2015 to limit to books less than 10 years old, and it indeed seems that 0based has become the norm among more recent books. Mathnerd314159 (talk) 20:47, 19 February 2024 (UTC)
 To quote page 17 of the Princeton Companion to Mathematics of 2008, "The natural numbers, otherwise known as the positive integers are ... 1, 2, 3, 4, and so on. ... (Some mathematicians prefer to include 0 as a natural number as well: for instance, this is the usual convention in logic and set theory. Both conventions are to be found in this book, but it should always be clear which one is being used.)" I guess that the editors were unable to get the contributors to the companion to agree to use one convention or the other. JonH (talk) 19:43, 19 February 2024 (UTC)
 So to illustrate what I mean by "removing 1 from the lead", it would be something like this: "In mathematics, the natural numbers are the numbers 0, 1, 2, 3, etc., also called the nonnegative integers. Outside of mainstream mathematics, there exists a differing interpretation among some individuals, who define the natural numbers to encompass the positive integers (1, 2, 3, etc.)." Mathnerd314159 (talk) 17:43, 19 February 2024 (UTC)
 There are two approaches to , the analytic (constructive) and the synthetic (axiomatic). In the analytic approach, the text defines naturals in terms of, e.g., sets, and it is extremely common to define the first natural as the empty set. In the axiomatic approach, starting from 0 and 1 are both common, but the definition of addition is slightly more convenient when starting from 1.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 11:08, 20 February 2024 (UTC)
The older source I was thinking of in my post above was Peano's Formulario mathematico. I just discovered, by reading the Peano article in Wikipedia, that: "Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number, while the axioms in Formulario mathematico include zero." The translation I read in college years ago was based on Formulario mathematico. I find it interesting to learn that not even the person who coined the word Natural Numbers had just one definition.
This is not a question Wikipedia can decide. We can only state the two different definitions. If someday mathematicians agree on a consensus, we will be able to state that. Unless that happens, there is no point in arguing about it here. Rick Norwood (talk) 11:36, 20 February 2024 (UTC)
 OK, this is an interesting aside. This is the first I've heard it claimed that Peano coined the term "natural number". Neither this article nor our bio of Peano seems to mention it. Where did you come across this claim? It seems unlikely to me; I would have thought it was much older. But I guess I don't really know that. Trovatore (talk) 17:43, 20 February 2024 (UTC)
 Unless there is an identical consensus for the axiomatic and constructive approach, the article would still need to mention both. I suspect that the hypothetical consensus will be 1 for axiomatic and 0 for constructive, but that guess is worth what you paid for it.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 14:36, 20 February 2024 (UTC)
 The issue is not stating the definitions. That is basic WP:NPOV that if there are differing viewpoints they should all be mentioned. The issue is giving WP:UNDUE weight to one definition vs. another, using a WP:IMPARTIAL tone and avoiding a WP:FALSEBALANCE of definitions that are most commonly used in very different contexts. By my count 34 of the past 50 edits (23 of them by @Montgomery Link who has so far not participated in this discussion) have been about changing the current 1biased phrasing to something giving 0based definitions more weight. So far they have all been reverted, but I don't think this situation is sustainable. The purpose of this discussion as I see it is to come to a WP:CONSENSUS on what the phrasing should be and how much weight to give one definition vs. the other (e.g., should the definition be in the first sentence, in the lead, in a section of the article; what definition should be used in each section; and so on). Mathnerd314159 (talk) 20:44, 20 February 2024 (UTC)
 I credited Peano. I should have also credited Pierce and Dedikind. There is, of course, a big difference between knowing about the Natural Numbers and naming them that. "The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano;" Rick Norwood (talk) 11:07, 21 February 2024 (UTC)
 There's also a big difference between giving them a name and giving them formal definitions. That quote doesn't say that any of those were responsible for the naming. My bet would be that the name goes back (in translation obviously) to the ancient Greeks. Trovatore (talk) 16:38, 21 February 2024 (UTC)
 What's in Book VII of Euclid's Elements?  Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:18, 21 February 2024 (UTC)
 The definition in Elements has (in translation): "A number is a multitude composed of units" with 1 as a "unit", so that the numbers begin from 2. However, in practice there are several places in the text where 1 is treated as a number. –jacobolus (t) 01:18, 4 June 2024 (UTC)
 Per MacTutor, who cites the OED (which is paywalled for me), the earliest attested use of "natural number" is Emerson in 1763. And then it is in the 1771 Encyclopaedia Britannica so presumably entered common usage from there. So not the ancient Greeks at all, Euclid just called them "numbers" [8]. The concept of "natural number" requires the concept of "unnatural numbers" as a contrast. Mathnerd314159 (talk) 18:52, 21 February 2024 (UTC)
 OK thanks. So not ancient, but still nearly a century before Peano was born, then. Might be worth working this into the article somehow. Trovatore (talk) 19:02, 21 February 2024 (UTC)
 What's in Book VII of Euclid's Elements?  Shmuel (Seymour J.) Metz Username:Chatul (talk) 18:18, 21 February 2024 (UTC)
 There's also a big difference between giving them a name and giving them formal definitions. That quote doesn't say that any of those were responsible for the naming. My bet would be that the name goes back (in translation obviously) to the ancient Greeks. Trovatore (talk) 16:38, 21 February 2024 (UTC)
 I credited Peano. I should have also credited Pierce and Dedikind. There is, of course, a big difference between knowing about the Natural Numbers and naming them that. "The second class of definitions was introduced by Charles Sanders Peirce, refined by Richard Dedekind, and further explored by Giuseppe Peano;" Rick Norwood (talk) 11:07, 21 February 2024 (UTC)
 Dedekind constructs the natural numbers in part V of Was sind und was sollen die Zahlen? (1888) and gives the first axiomatization there. Peano credits him for this in his (1889). Enderton makes the following relevant comment in a footnote on p. 66 of Elements of Set Theory (Academic Press, 1977): "Is 0 a natural number? With surprising consistency, the present usage is for school books (through highschool level) to exclude 0 from the natural numbers, and for upperdivision collegelevel books to include 0. Freshmen and sophomore college books are in the transition zone." If so, then part of what is at stake is the intended audience for this article. Montgomery Link (talk) 01:20, 17 March 2024 (UTC)
 That source seems sufficiently neutral, I think it is about as good as it gets as far as characterizing perspectives. I have updated the lead. Mathnerd314159 (talk) 17:29, 18 March 2024 (UTC)
 As far as audience, you cannot really assume anything about the reader's level. There is some discussion in WP:TECHNICAL#Audience but with an article like this it can be assumed that both primary school kids and expert mathematicians will be reading it. In this case I think phrasing it as college vs. elementary school is sufficiently clear that a kid can complain "that definition is not what this advanced math textbook says" and the elementary school teacher can say "... sorry, that's what our textbook says." Mathnerd314159 (talk) 17:48, 18 March 2024 (UTC)
 I'm skeptical that it's a good idea to go into this sort of detail about when 0 is or is not included, especially based on a single source, even as reputable as Enderton. The language about "high school" and "college" suggests that he may have been thinking in terms of the US when he wrote this (and even if he wasn't, if we use these terms, it sounds like we are). I think it's likely better to say sometimes it is included, and then again sometimes it isn't, and pretty much leave it at that. Trovatore (talk) 18:20, 18 March 2024 (UTC)
 Well, the available evidence (Enderton, Ztimes3's comment above, my searches of Google Books and Scholar, most of the discussions, looking back) suggests this particular split in usage patterns is enduring and international. If you have more universally accepted descriptors to use in the lead, go ahead, but I think "modern college and professional mathematics" vs "elementary education and classic or niche mathematical texts" is a detail that is relevant for most readers, particularly ones wondering which convention is in use when they encounter a source that does not make it explicit. Although checking a comparison it seems "elementary" is more of an American term, I think other English speakers will have no problem understanding it. I guess changing "college" to "university" might be appropriate though. Mathnerd314159 (talk) 20:04, 18 March 2024 (UTC)
 I agree with Trovatore that we must say simply that 0 is sometimes included and sometimes not. If we try to classify the the usage by the mathematical level, you will imply that one should not use a sentence such as "the rows and columns of a matrix are commonly labelled with the first natural numbers". In fact, the choice of including 0 or not depends mainly of convenience in each mathematical context. D.Lazard (talk) 20:42, 18 March 2024 (UTC)
 Looking at Matrix (mathematics)#Notation, there is a note "Some programming languages start the numbering of array indexes at zero [...] This article follows the more common convention in mathematical writing where enumeration starts from 1." So in this case using "natural number" is perfect, as it leaves ambiguous whether it starts from zero or one. The fact that 1 is more common in this case falls into my wording of "niche mathematics"  subscripts are indeed a niche distinct from more common uses for natural numbers.
 I don't think we can simply say "0 is sometimes included". Per WP:NPOV, the article must "indicate the relative prominence of opposing views." A basic statement that differing conventions exist doesn't do this. In this case, there is a specific source (Enderton) that explains the relative prominence of the definitions. I would think about writing "Enderton says that...", but then there is also "Avoid stating facts as opinions", and it is clear from factchecking Enderton's statement that it is a fact, so instead the relative prominence is described in Wikipedia's POV. Mathnerd314159 (talk) 21:08, 18 March 2024 (UTC)
 I'm quite sure that quote from Enderton is wrong, although I can understand how he would make that mistake, since practically all set theorists and set theoryadjacent people take 0 ∈ N. For example, I think most number theorists take N to start at 1, e.g. see p. 9 of Iwaniec 2004 "Analytic Number Theory" (this is certainly not a "freshman" or "sophomore" textbook) — this is just the first advanced number theory textbook I could find that clearly defined the natural numbers. Popcountll (talk) 22:50, 18 March 2024 (UTC)
 It does seem convincing, all the number theory books I looked at either avoided the term or started at 1. Would you accept "In most modern college and professional mathematics, the natural numbers are the numbers 0, 1, 2, 3, ..., also called the nonnegative integers. In number theory, elementary education, and in classic or niche mathematical texts, it is common to instead define the natural numbers as excluding zero, corresponding to the positive integers 1, 2, 3, ..."? Mathnerd314159 (talk) 23:59, 18 March 2024 (UTC)
 I just don't see any need to go into that level of detail. It's like articles that start with large digressions about what varieties of English use one word or spelling and which ones use another one. It's a distraction from the main thrust of the article. Trovatore (talk) 00:52, 19 March 2024 (UTC)
 I believe that the article should go into that level of detail, but not the lead.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:38, 19 March 2024 (UTC)
 That would be less problematic, but I'm still skeptical that we can cover this division in a justifiable and sourceable way. The passage from Enderton is exactly the sort of thing we would need, but unfortunately it appears to be inaccurate, or at least not accurate in the full generality we'd want. On the other hand, the sort of analysis where you look at various texts and say, ah hah, these numbertheory texts start with 1 and these settheory texts start with 0, so we'll say it starts with 1 in number theory and 0 in set theory, is the kind of original research we can't do here.
 I just honestly don't see the point of this sort of analysis. For the most part, the things that we want to say about the natural numbers don't depend on whether zero is included. We do have to cover the different conventions, but I think after stating what they are, the less said the better. Trovatore (talk) 17:57, 19 March 2024 (UTC)
 There is a line in the WP:NPOV policy, "This policy is nonnegotiable, and the principles upon which it is based cannot be superseded by other policies or guidelines, nor by editor consensus." I would presume that those "other policies" include verifiability and no original research  apparently it is more important to create an analysis of the relative prominence of views than it is to justify or source that analysis. To paraphrase from WP:RNPOV (emphasis mine): "NPOV policy means Wikipedia editors ought to try to write sentences like this: "Certain mathematicians (such as Enderton) believe natural numbers start from 0 and consider this definition to be standard for universitylevel mathematics. Other mathematicians who call themselves number theorists—influenced by the assumptions required for certain theorems (such as Van der Waerden's theorem)—instead believe that defining the natural numbers as starting from 1 is more appropriate." Mathnerd314159 (talk) 23:13, 19 March 2024 (UTC)
 Mathnerd, you seem to have the idea that this is an actual dispute that we need to figure out how to cover with due weight. That's just not so. There is no dispute. There are different conventions. You're taking it way too seriously. Trovatore (talk) 23:21, 19 March 2024 (UTC)
 Saying "0 is sometimes included and sometimes excluded" is clearly conform to the policy WP:NPOV, and this assertion can be sourced by Enderson's book. On the other hand, Enderson's analysis of the cases where 0 is included or not is WP:OR, since it is a primary source, and we do not have any secondary source discussing it. So, both WP:NPOV and WP:NOR imply that Enderson's asssertion that "0 is not included at elementary level and 0 is included at higher level" must either be omitted or explicitly attribued (Enderson wrote: "..."). If not omitted, WP:NPOV imply that such an opinion does not belong to the lead. D.Lazard (talk) 09:35, 20 March 2024 (UTC)
 There is a line in the WP:NPOV policy, "This policy is nonnegotiable, and the principles upon which it is based cannot be superseded by other policies or guidelines, nor by editor consensus." I would presume that those "other policies" include verifiability and no original research  apparently it is more important to create an analysis of the relative prominence of views than it is to justify or source that analysis. To paraphrase from WP:RNPOV (emphasis mine): "NPOV policy means Wikipedia editors ought to try to write sentences like this: "Certain mathematicians (such as Enderton) believe natural numbers start from 0 and consider this definition to be standard for universitylevel mathematics. Other mathematicians who call themselves number theorists—influenced by the assumptions required for certain theorems (such as Van der Waerden's theorem)—instead believe that defining the natural numbers as starting from 1 is more appropriate." Mathnerd314159 (talk) 23:13, 19 March 2024 (UTC)
 I believe that the article should go into that level of detail, but not the lead.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:38, 19 March 2024 (UTC)
 I just don't see any need to go into that level of detail. It's like articles that start with large digressions about what varieties of English use one word or spelling and which ones use another one. It's a distraction from the main thrust of the article. Trovatore (talk) 00:52, 19 March 2024 (UTC)
 It does seem convincing, all the number theory books I looked at either avoided the term or started at 1. Would you accept "In most modern college and professional mathematics, the natural numbers are the numbers 0, 1, 2, 3, ..., also called the nonnegative integers. In number theory, elementary education, and in classic or niche mathematical texts, it is common to instead define the natural numbers as excluding zero, corresponding to the positive integers 1, 2, 3, ..."? Mathnerd314159 (talk) 23:59, 18 March 2024 (UTC)
 I'm quite sure that quote from Enderton is wrong, although I can understand how he would make that mistake, since practically all set theorists and set theoryadjacent people take 0 ∈ N. For example, I think most number theorists take N to start at 1, e.g. see p. 9 of Iwaniec 2004 "Analytic Number Theory" (this is certainly not a "freshman" or "sophomore" textbook) — this is just the first advanced number theory textbook I could find that clearly defined the natural numbers. Popcountll (talk) 22:50, 18 March 2024 (UTC)
 I agree with Trovatore that we must say simply that 0 is sometimes included and sometimes not. If we try to classify the the usage by the mathematical level, you will imply that one should not use a sentence such as "the rows and columns of a matrix are commonly labelled with the first natural numbers". In fact, the choice of including 0 or not depends mainly of convenience in each mathematical context. D.Lazard (talk) 20:42, 18 March 2024 (UTC)
 Well, the available evidence (Enderton, Ztimes3's comment above, my searches of Google Books and Scholar, most of the discussions, looking back) suggests this particular split in usage patterns is enduring and international. If you have more universally accepted descriptors to use in the lead, go ahead, but I think "modern college and professional mathematics" vs "elementary education and classic or niche mathematical texts" is a detail that is relevant for most readers, particularly ones wondering which convention is in use when they encounter a source that does not make it explicit. Although checking a comparison it seems "elementary" is more of an American term, I think other English speakers will have no problem understanding it. I guess changing "college" to "university" might be appropriate though. Mathnerd314159 (talk) 20:04, 18 March 2024 (UTC)
 @Montgomery Link Just thought I'd translate the title of the Dedekind book you mentioned for anyone interested: “What are and what should the numbers (do)” or, less literally “What are and what are the numbers for?”.
 I'm skeptical that it's a good idea to go into this sort of detail about when 0 is or is not included, especially based on a single source, even as reputable as Enderton. The language about "high school" and "college" suggests that he may have been thinking in terms of the US when he wrote this (and even if he wasn't, if we use these terms, it sounds like we are). I think it's likely better to say sometimes it is included, and then again sometimes it isn't, and pretty much leave it at that. Trovatore (talk) 18:20, 18 March 2024 (UTC)
 SaintIX (talk) 23:39, 4 June 2024 (UTC)
 Thanks. You might be interested in a noted translation of the same, which is to be found in Ewald's From Kant to Hilbert : a source book in the foundations of mathematics, vol. 2.
 My understanding is that zero was introduced historically as a place holder (see, e.g., section 1 of https://www.jstor.org/stable/pdf/3109881.pdf). See fn 10 in the same for evidence that Dedekind was including 0 as the empty set sometime around 1887, and clearly so by the 1890's; so there is the blunt issue with obvious ramifications. There in section 1 it also says that Frege needs zero for logical reasons, but Dedekind uses one as his "base". The map from {0, 1, 2, ...} to {1, 2, 3, ...} is oneone and onto (pf. uses the successor function), so the issue seems somewhat arbitrary. And yet, one point that no one has raised is the notation issue and the history of zero. To exclude zero from the natural numbers is to make them notationally inadequate. What is natural about that? Montgomery Link (talk) 00:37, 5 June 2024 (UTC)
 @Montgomery Link Thank you. German is actually my first language, although my English is much better than my German by now (I was 9½ when we moved to England in Jan '69 and had spent a year here – Aug '66Jul '67 – and one in the USA – Aug '67Jul '68; academic years: my father partook in exchange lectureships). I do still read German when that's the original language of novels but technical books might present me some difficulty, so the suggestion for the translation is appreciated 🙏🏼. SaintIX (talk) 01:12, 5 June 2024 (UTC)
Alright. So then I guess there are two changes. (1) Use the "possibly excluding 0" wording in the lead. D. Lazard supported this, and the slight bias towards 0 can be justified by the ISO standard and/or Enderton (and that any wording which starts with an example will be biased towards that example). Also, there are cases like "all natural numbers are prime or composite" where what is meant is to not only exclude 0 but also 1 (similar to Euclid distinguished multitudes and units). With a 1biased definition the tendency of excluding special cases is less clear.
(2) Expand the "Modern definition" section to discuss the differing conventions and their use. There is the origin of the term "natural number" (likely Emerson per OED), Peano using the letter N and switching from 1 to 0, and then Enderson's opinion and some piecemeal citations of prominent textbooks. Probably also a reference to the New Math movement if I can find a source. Mathnerd314159 (talk) 04:20, 7 April 2024 (UTC)
 We can argue until the cows come home, but unless we appoint a dictator of mathematics, it is all just words. I think 1, 2, 3, ... . You think 0, 1 2, 3, ... . i teach that ℕ = {1, 2, 3, ... } but I tell my students about the difference of opinion. And, of course, it is possible to define ℕ one way and "the natural numbers" the other way. But the main point is that, of all the ways of deciding this question, deciding it on Wikipedia is probably the worst.
 Moo! Rick Norwood (talk) 17:34, 7 April 2024 (UTC)
 guys, if we consider group and ring theory, isn't having 0 as a part of natural numbers makes much more sense? Like it follows the property, and without it, then we'll have issues. So why are we still arguing, aren't we suppose to follow logic? Also computer scientist will have issues with how it begins from 1.(also if you use padic numbers, the first number is still 0) most of not all auxiliary notions points towards 0 being essential in the group. I still don't understand how someone claim that most number theorist consider 0 to not be in natural numbers, when all the other groups needed it. Please consider multiple angles, from set theory, to padic and all angles, not your feeling of what should be, we are mathematicians, and if you are citing me sources, considered also if they are objective or subjective, because evidences from different fields are pointing to include 0 into natural numbers, humbly, New1997.— Preceding unsigned comment added by 183.182.115.217 (talk • contribs) 00:51, 4 June 2024 (UTC)
 Regarding number theory, there is the book "From Great Discoveries in Number Theory to Applications", and p. 9 of Iwaniec 2004 "Analytic Number Theory" from Popcountll above, and my own cursory search on Google Books. I guess it is not strictly cited in the article but I don't really want to throw in a citation of 100 number theory books just to prove a point. In group and ring theory (algebra, as cited by Gerald A. Edgar by MacTutor), the textbooks mostly do include 0 as a part of natural numbers. And similarly logic and set theory and CS.
 Regarding multiple angles, that is sort of the point, one of my earlier attempts was "the natural numbers are the numbers 0, 1, 2, 3, ..." and that was reverted. Regardless of anyone's personal feelings, it is simply not accurate to say that one definition is correct and one is wrong. The evidence I pulled up shows the term has been ambiguous since the 1700s, and if ISO adopting one definition as a standard is not sufficient to change that I am not certain what is.
 From your comment though, it does seem though that the current state of affairs is still not satisfactory. I honestly am not sure what is left to try besides my earlier suggestion of splitting into positive integer and nonnegative integer and making this page a disambiguation or broad concept. WP:NOTDICTIONARY says "Articles almost always focus on a single definition or usage of the title", which does sort of justify it. It is fairly clear in the current article which parts are 0based and which are 1based, so splitting it would not be hard. Thoughts, anyone? Mathnerd314159 (talk) 05:04, 4 June 2024 (UTC)
 edit: new section Mathnerd314159 (talk) 05:19, 4 June 2024 (UTC)
 @Mathnerd314159 I agree with you, although, as both ℕ₀ and ℕ₁ are definitions of the Natural numbers, I'm not sure what splitting them would achieve? As far as set theory goes, the Natural numbers are a bit problematic anyway, at least under addition: without zero there is no identity element and there are no inverse elements without including ℤ, the negative integers. I hope you'll excuse me, it's been a while since I did my degree (19936, in Computer Science & Discrete Mathematics) and I don't often get the opportunity to practise, much less discuss, any maths. My father is a retired mathematician and his reaction to the question whether zero is a natural number was “Of course it is!” (his demeanor was almost how could you ask such a stupid question 😄). Purely personally, I tend to the opposite view, that zero isn't a ‘natural’ number. I'd go further: zero is actually quite a sophisticated and relatively recent concept (its introduction into European mathematics is attributed to Fibonacci in the 10ᵗʰ11ᵗʰ Century; qv Zero) and many ancient numbering systems, eg the Roman and Greek, didn't include it all. In fact the Greeks argued about it quite a lot, both from a philosophical and later a religious standpoint. To play devil's advocate, I'm tempted to propose that zero isn't actually a number at all, much as black isn't strictly speaking a colour (in the same way that black is the absence of colour, one could argue that zero is the absence of number. As I understand it, without further research, that seems to be at the root of the Greeks' arguments about it: how can nothing be a number?). However, I'm even less qualified to debate philosophy than I am mathematics. For what it's worth and despite being against my instincts, better judgement and feelings (none of which are, admittedly, factual or scientific), if it really is imperative that we plump for one definition over another, I'd vote for ℕ₀ (as opposed to ℕ₁, it being the ISO definition and the one most mathematicians subscribe to). Having said that, my actual inclination is to let things stand as they are. Not withstanding the WP:NOTDICTIONARY guidelines (although I would point out that they do say “…almost always…” and this may well be a prime example of where it's neither appropriate nor desirable to have a single definition), pointing out the ambiguity gives a more complete and accurate definition. As illustrated by the discussion here and the lack of unanimity even amongst mathematicians, it's actually a surprisingly contentious and thorny issue, hence my inclination to let sleeping dogs lie. Splitting the article may just open a whole new can of worms (please excuse the clichés, I couldn't resist them 😉). I imagine it would also lead to some confusion for Wikipedia users looking for a definition.
 Just out of curiosity, if it were decided to split the article into two, what would you suggest for their headings? Would it be fair or even accurate to have one described as the ‘Natural Numbers’ and the other not, although, surely at least one of them should have that heading?
 In conclusion I'd just like to say that while I'm earnest about the points I've made, my tone is intended to be lighthearted, something that it's not always easy to convey in writing, but I do hope you don't find what I've said too frivolous. I sincerely value and enjoy being a part of this community, however minor my contributions may be, and I'd like to thank you all for taking the time to read this far and allowing me that privilege 🙏🏼. SaintIX (talk) 23:20, 4 June 2024 (UTC)
 Splitting would put different definitions on different articles, as required by WP:NOTDICT. There is no debate about what the "positive integers" or "nonnegative integers" are, so using those as the titles would end the argument. A fuzzy concept like "zero is not a number" is made clear: it is a nonnegative integer but it is not a positive integer. As far as votes, typically the only options in a vote are support or oppose, but I suppose you could cast a vote to define the natural numbers to only be ℕ₀ in the split proposal below. It sounds like you otherwise oppose the split because it is a "can of worms" but I am not clear exactly why it would be worse than the current situation. Mathnerd314159 (talk) 02:45, 5 June 2024 (UTC)
black isn't strictly speaking a colour
– as an offtopic aside, black is most certainly a color by any plausible scientific definition of "color". –jacobolus (t) 07:20, 5 June 2024 (UTC) @Jacobolus Please don't take offence but I have to disagree: the scientific definition of colour is “…the visual perception based on the electromagnetic spectrum” (to quote Wikipedia's entry on colour) and black isn't on the electromagnetic spectrum, so certainly not any plausible definition. Mind you, I would agree that that definition isn't entirely satisfactory as there is no such thing as brown light and I don't think anyone would argue that brown isn't a colour. The problem there lies with additive (light) as opposed to subtractive (pigments) colouring – the primary colours of the former, eg in your TV screen, are RGB whereas in print the primary colours are CMYK (cyan, magenta, yellow & ‘key’ – key being black. It's called key because the keyline, used for outlines of solid blocks of colour was usually part of the black plate. Theoretically, a combination of CMY should give you black but there are a number of problems using them: the inks aren't usually pure enough and actually give a ‘muddy’ black; using all three saturates the paper causing it to warp or even tear; black is cheaper than the other colours so it costs less than ⅓ as much).
 I agree with you in that normally I would call black a colour but it comes down to semantics (how DO you define colour?), which was really my point in the first place (if a number is a quantity, how can nothing be a number? Nothing may be ‘a thing’ but is it something? If zero is a ‘natural’ number why did it take so long to be commonly accepted as a number rather than a placeholder? I'm not looking for answers, they're rhetorical, to illustrate what I mean by it being a matter of semantics and/or philosophy. Nevertheless, I do find them interesting 🤔, sorry).
 Just as a further aside, black pigments are actually rarely really black, if ever. When I had my bass drum refinished in black and compared it to my tomtoms, the toms had a slight green tint whereas the bass drum had a slight red tint. The difference was only noticeable under very strong light and without having them next to one another I would have called them both black. Two finishes, both sold as black, one from the factory, the other from a small instrument maker.
 Please accept my apologies for going so far offtopic, it's just that I do enjoy these discussions. This is a ‘talk’ page, after all, so I did not think it entirely inappropriate. SaintIX (talk) 20:15, 6 June 2024 (UTC)
 The Wikipedia entry doesn't really give the most accurate definition, and as a result you are misinterpreting it. If you go look in any color science book, or in the definitions used by e.g. the CIE, you will find that the key part of the definition of "color" is human perception, not light spectra or physical properties of objects. Beyond that, as you point out, the word "black" applies to a wide range of perceptions. –jacobolus (t) 20:30, 6 June 2024 (UTC)
 @Jacobolus Thank you for that. And I totally agree. If asked I would have said colour is the quality we ascribe to our perception of certain wavelengths of the electromagnetic spectrum. I think perception is the key word there and, ultimately, those are subjective. Perhaps someone should edit the entry on colour 😉? Anyway, thanks again. I'm always flattered when someone takes the time to reply to me ☺️. SaintIX (talk) 21:06, 6 June 2024 (UTC)
 The Wikipedia entry doesn't really give the most accurate definition, and as a result you are misinterpreting it. If you go look in any color science book, or in the definitions used by e.g. the CIE, you will find that the key part of the definition of "color" is human perception, not light spectra or physical properties of objects. Beyond that, as you point out, the word "black" applies to a wide range of perceptions. –jacobolus (t) 20:30, 6 June 2024 (UTC)
 Actually, there 'is a debate about what the "positive integers" are; some authors use it to mean nonnegative and use the term "strictly positive" when the want to exclude zero.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 12:36, 7 June 2024 (UTC)
 @Chatul That's pretty much what I've discovered as I've delved into this subject (more than I really wanted or intended to). The argument goes back thousands of years and I'm not arrogant or hubristic enough to think we'll solve it here.
 We are, kind of, back to language and that's tricky at best, as anyone bilingual will know… SaintIX (talk) 13:06, 7 June 2024 (UTC)
 I do not think the splitting idea is a good one. I remember many years ago when I heard that this controversy existed, but I never personally ran into it until now, but I see why I was told it was heated. In the mathematical logic scene around the world, zero is included in ℕ. In the Hartley Rogers classic Theory of Recursive Functions and Effective Computability he calls ℕ (this is the first symbol he introduces in the book) the class of "integers" and includes zero. He means the nonnegative integers. Alonzo Church in the classic Introduction to Mathematical Logic includes zero (p. 179 and see the interesting fn 521). Montgomery Link (talk) 00:59, 5 June 2024 (UTC)
 @Montgomery Link I agree with both points: that splitting isn't a good idea and that zero is an element of ℕ, certainly mathematically. The views I expressed elsewhere in this talk we really more philosophical than mathematical, more to do with the concept of zero and whether it can be regarded as a ‘natural’ number or even a number at all. I guess part of the controversy is actually more a matter of semantics and whether ‘natural’ is really a good name for ℕ. However, that's not a subject for Wikipedia, much less an article on ℕ and given its wide use, as well as its symbol, it's something we're stuck with. Mind you, ℤ isn't that obvious unless you know that it stands for ‘Zahl’ or ‘Zahlen’ (German for ‘Number/s’) nor is ℚ, the rational numbers, unless you know it stands for ‘Quotients’.
 Excuse the (slightly pedantic) explanations, they're intended for the benefit of people who aren't as knowledgeable as you. I certainly didn't intend to impugn your understanding. SaintIX (talk) 01:44, 5 June 2024 (UTC)
Split proposal
[edit]The following discussion is closed. Please do not modify it. Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.
The debate about the meaning of the natural numbers has gone on for way too long. Therefore, I propose that the article be split into two separate pages called Positive integer and Nonnegative integer (currently both redirects to here). The content will be split among them according to whether it uses the 0based or 1based definition of natural number, with links between the pages as appropriate. This page will be a broad concept article dicussing usage of the term "natural numbers". There are sufficient references discussing its ambiguity and definitions to establish notability.
As justification, there is WP:NOTDICT. Under criteria, it lists that the proper resolution of "the same title for different things (homographs)" is to have different articles for the different things and then a disambiguation page. This is clearly the case, the natural numbers is a title used for both the positive integers and the nonnegative integers. Mathnerd314159 (talk) 05:19, 4 June 2024 (UTC)
 Strongest possible oppose. Come on, this is just silly. Whether zero is included or not is a triviality, not an indication that we're talking about a whole different subject matter. As I said in an earlier discussion, very little that we want to say about the natural numbers changes in any substantial way depending on whether or not zero is included. Trovatore (talk) 07:33, 4 June 2024 (UTC)
 Precise language and definitions can be a matter of life and death. Not sure there are any specific examples of zero vs. onebased, as it typically is caught early in testing, but Ariane flight V88 is an example of the general gravity of the situation. It is not "silly" or "a triviality". Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 I'm not going to engage in these sophomoric quibbles. If you're willing to take it seriously I'll elaborate more, but not in response to this sort of fluff. Trovatore (talk) 22:14, 4 June 2024 (UTC)
 Precise language and definitions can be a matter of life and death. Not sure there are any specific examples of zero vs. onebased, as it typically is caught early in testing, but Ariane flight V88 is an example of the general gravity of the situation. It is not "silly" or "a triviality". Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 Oppose. Per Trovatore, and because this does not solve anything: as "natural number" is a common search phrase, we must keep it as a redirect, and here is no way for choosing its target. A dab page would not be a good solution either, since most reader do not care whether 0 is included or not. D.Lazard (talk) 09:01, 4 June 2024 (UTC)
 In popularity on Wikipedia, "natural number" is about the same pageviews as integer, number, real number, complex number: [9]. WikiNav says 48% of traffic is from search. In Google Trends, it is maybe 9060201 for whole number / natural number / positive integer / nonnegative integer. [10]. Certainly this would argue that "nonnegative integer" is a bad page title, but there is no more common term that unambiguously describes the integer 0, 1, 2 etc. And if popularity was everything then whole number would not be a disambiguation. If we include "natural number" in the nonnegative integer page then Google will most likely pick up on it and give it as the top result for "natural number". Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 Wikipedia naming is not a strict popularity contest.
 What we have here is two essentially identical concepts with an offbyone error in their (arbitrary) conventions. The names "whole number" and "counting number" are usually reserved for children, laypeople, and very informal technical conversations, and "whole number" is furthermore ambiguous because it often includes negative integers. Neither of them is an appropriate title here in my opinion.
 The names "nonnegative integer" and "positive integer" needlessly exclude the other set, which should be discussed in the same place.
 There's no problem at all with using "natural number" as a title for this article. This discussion is a tempest in a teapot. The longstanding stable version of this article was completely fine. The article shouldn't belabor this point in the first few sentences, and doesn't need to stake a claim in the fight. It just needs to briefly mention the difference of convention in the lead, and then discuss any practical effects of the choice somewhere later down the page. The exact phrasing used is up for discussion and should be settled based on consensus, but isn't really that big a deal. –jacobolus (t) 02:36, 5 June 2024 (UTC)
 "The names 'nonnegative integer' and 'positive integer' [...] should be discussed in the same place."  I don't agree with this. There is nothing useful that can simultaneously be said about 0,1,... and 1,2,... that would not also apply to more general concepts such as an integer sequence. Whereas, there are many things that can be said specifically about one or the other, such as the history of counting and the correspondence with the cardinal or ordinal numbers. Mathnerd314159 (talk) 06:48, 5 June 2024 (UTC)
 In popularity on Wikipedia, "natural number" is about the same pageviews as integer, number, real number, complex number: [9]. WikiNav says 48% of traffic is from search. In Google Trends, it is maybe 9060201 for whole number / natural number / positive integer / nonnegative integer. [10]. Certainly this would argue that "nonnegative integer" is a bad page title, but there is no more common term that unambiguously describes the integer 0, 1, 2 etc. And if popularity was everything then whole number would not be a disambiguation. If we include "natural number" in the nonnegative integer page then Google will most likely pick up on it and give it as the top result for "natural number". Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 Oppose Mathnerd314159, you can't be serious with this one? –jacobolus (t) 14:49, 4 June 2024 (UTC)
 It is definitely the "nuclear" option, but as I work out the details it seems more and more a good idea. Nobody has argued with the central thesis, WP:NOTDICT, which is established Wikipedia policy. Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 It seems to me like you are mad that people disagree with you and are flailing a bit. –jacobolus (t) 22:22, 4 June 2024 (UTC)
 I am not particularly mad. I would say more disappointed that nothing has worked so far. I have tried all of the other solutions mentioned in my post of 00:24, 19 February 2024 (UTC), besides splitting. To paraphrase Sherlock Holmes, "When you have eliminated the impossible, whatever remains, however improbable, must be the answer." I am not saying the current state is bad, but it also is still generating discussion, so it is not good enough. Mathnerd314159 (talk) 06:57, 5 June 2024 (UTC)
 It seems to me like you are mad that people disagree with you and are flailing a bit. –jacobolus (t) 22:22, 4 June 2024 (UTC)
 It is definitely the "nuclear" option, but as I work out the details it seems more and more a good idea. Nobody has argued with the central thesis, WP:NOTDICT, which is established Wikipedia policy. Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 OPPOSE The two systems of natural numbers are naturally isomorphic and are often described in terms of the same Peano postulates. From a mathematical perspective it makes no sense to treat them differently.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 17:32, 4 June 2024 (UTC)
 You say they are "the same", but then they are written , , clearly different. Under the Banach–Tarski paradox one ball is isomorphic to two balls, are you going to make 2 redirect to 1 because "there is no mathematical difference"? Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 Natural numbers form a single concept, with two slightly different definitions. So WP:NOTDICT does not apply. The fact that the definitions are very similar can be seen on Peano axioms, the standard way to formally define the natural numbers: one passes form one definition to the other by changing the axiom "0 is a natural number" into "1 is a natural number" (and vice versa), and keeping the other axioms unchanged. Moreover, many mathematical texts do not state clearly which definition they use, generally because this does not matter. So a dab page (that you suggest implicitely) would be confusing, since most readers would not know which link to choose. So, it is much clearer to have a single article that states clearly that there are two variants in the definition. D.Lazard (talk) 21:22, 4 June 2024 (UTC)
 I agree. Chatul has made a strong case for not splitting. Montgomery Link (talk) 01:03, 5 June 2024 (UTC)
 "Natural numbers form a single concept"  citation needed. Certainly concept is a very vague term, but one definition of concept on Google is "a theoretical construct within some theory" and within set theory it is clear that ℕ₀ and ℕ₁ are different sets and therefore different concepts. Similarly the current definition "the numbers 0, 1, 2, 3, etc., possibly excluding 0" is deliberately vague.
 Regarding mathematical texts being lazy, this is a perennial problem. Some have learned but often besides lazy definitions there are glaring errors. All it means is that they are not reliable sources and not relevant to the argument. You cannot use a pile of dung to paint a masterpiece. (unless you are Chris Ofili) As far as "it doesn't matter", citation needed, again. I think it is more likely that students have wasted countless hours searching for clues to figure out which definition is meant.
 "a dab page would be confusing"  I didn't propose this, I proposed a broad concept page. The content would be the current lead, suitably modified, plus the discussion of natural numbers as a term. But even if we did end up with a dab page, it would not be any worse than whole number. If you are going to wage a crusade against confusing dab pages you should start there. (My proposal to turn it into a broadconcept page that actually explained when it was used as integer vs. positive integer vs. nonnegative integer was opposed by Trovatore)
 "it is much clearer to have a single article"  here we get into WP:MERGE. It is certainly debatable but there is WP:NOTMERGE, "Merging should be avoided if: ... The topics are discrete subjects warranting their own articles, with each meeting the General Notability Guidelines, even if short." I think it is clear that the positive integers and the nonnegative integers are independent, discrete subjects and meet GNG. I also think Wikipedia would be clearer if it consistently used positive integer / nonnegative integer and deliberately avoided ever using the term "natural number". Having a single article combining two concepts encourages the sort of sloppy, imprecise mathematical writing you mention. Mathnerd314159 (talk) 03:22, 5 June 2024 (UTC)
 The "single concept" here is an ordered, countablyinfinite sequence, which can be used for enumerating other finite or countably infinite sequences, comparing their sizes, indexing finite or infinite sums, building up other kinds of numbers using various data structures with natural numbers as a basic type (or sometimes itself a constructed type), and so on. Depending on context sometimes it's convenient to start with one or the other (and occasionally even with some other starting point like or ), but the basic purpose and idea is the same in any event, and the exact starting point is largely irrelevant.
 We can quibble about whether an additive identity should be included or not, at what index value we should start counting any particular sequence, the exact details of the representation(s) when trying to represent arbitrary rational numbers using some structured collection of natural numbers, etc. However, people rarely use natural numbers per se for more general kinds of arithmetic since the full set of integers (or more capable supersets) is easily available and ready for arithmetic. –jacobolus (t) 05:56, 5 June 2024 (UTC)
 By "ordered, countablyinfinite sequence" I understand you to mean an integer sequence, specifically an arithmetic progression or a sequence of consecutive integers. That's three concepts. And none of these pages redirect to the natural numbers, or are linked here, so I conclude they are distinct concepts.
 "Depending on context sometimes it's convenient to start with [...] some other starting point like 1 or 3"  would starting with something else be called the natural numbers though? I think that is what you are missing, is that there are precisely two definitions of the natural numbers, the positive integer definition and the nonnegative integer definition. So a disambiguation page with two links is the most structured representation of the meaning of the natural numbers.
 And it is not "we can quibble", it is "we HAVE quibbled", for several centuries (and 20 years onwiki), over which of these two definitions is appropriate. The other things you mention such as rational numbers or arithmetic have not been an issue. Mathnerd314159 (talk) 06:38, 5 June 2024 (UTC)
 The Greeks (sometimes) started their equivalent list from 2. It didn't really make any fundamental difference. If you want you could start from the number . It's nothing more or less than a difference in convention, and is at worst mildly inconvenient to work around sometimes. The important part is to have a particular welldefined list that you can use as a basis for comparison, but where precisely it starts is a triviality.
 In your own textbook, please pick whichever one you prefer and stick to it. In Wikipedia, we should briefly mention that conventions differ (if only to forestall timewasting complaints from pedants), and then move on to discuss the actual subject: the use and structure of these tools. –jacobolus (t) 07:14, 5 June 2024 (UTC)
 "If you want you could start from the number 4,321"  no? I mean, if I change the natural numbers page to say "the natural numbers start from the number 4,321" it will clearly get reverted.
 "The important part is to have a particular welldefined list"  I agree, but since the natural numbers are not welldefined the only way to get a welldefined list is to avoid the term entirely and use more specific terms.
 "In your own textbook, please pick whichever one you prefer and stick to it."  I'm not writing a textbook. I'm here to build an encyclopedia.
 "In Wikipedia, we should briefly mention that conventions differ, and then move on to discuss the actual subject". That would be great, if it was possible. For example see countably infinite  there are similarly two different definitions. Unlike here, the article picks one definition and sticks with it. Whereas here, the history, notation, even some of the rest of the article is devoted to discussing the different conventions. Certainly it would be great if we could just say "the natural numbers start at 0" and discuss alternative conventions in a section, but that doesn't seem possible. Mathnerd314159 (talk) 18:19, 5 June 2024 (UTC)
 This discussion, for which the conclusion (i.e. more or less the previous stable version) is the overwhelming consensus of editors, is wasting people's time and is disruptive to the process of building an encyclopedia. Repeatedly misunderstanding and mischaracterizing several other editors' comments and arguments and then wikilawyering about the straw man version is not helping convince anyone. Instead of making increasingly implausibletohappen demands based on increasingly exaggerated arguments, it would be a dramatically better use of everyone's time if you either (a) give up, or (b) figure out where other editors might agree with one or another part of your goals, and then focus on small, deliberate changes that have some chance of attaining consensus. –jacobolus (t) 19:50, 5 June 2024 (UTC)
 The situation was not "the overwhelming consensus of editors", it was established by user Dissident in 2004. The choice to redirect Positive integer and Nonnegative integer here was never discussed, I checked. Silence is not consensus, or at least it is the weakest form and the presence of a disagreeing editor such as myself disrupts that consensus.
 Now it is true, if indeed we wait out the week and nobody besides me thinks splitting is a good idea, it is fair to conclude from the number of oppose votes here that there is a consensus for the current page structure. Mathnerd314159 (talk) 23:03, 5 June 2024 (UTC)
 No. Consensus does not require unanimity. Otherwise, no article could be deleted in the face of a single "keep" !vote, no user could ever be blocked in the face of a single "oppose" at ANI, etc. XOR'easter (talk) 00:09, 6 June 2024 (UTC)
 This discussion, for which the conclusion (i.e. more or less the previous stable version) is the overwhelming consensus of editors, is wasting people's time and is disruptive to the process of building an encyclopedia. Repeatedly misunderstanding and mischaracterizing several other editors' comments and arguments and then wikilawyering about the straw man version is not helping convince anyone. Instead of making increasingly implausibletohappen demands based on increasingly exaggerated arguments, it would be a dramatically better use of everyone's time if you either (a) give up, or (b) figure out where other editors might agree with one or another part of your goals, and then focus on small, deliberate changes that have some chance of attaining consensus. –jacobolus (t) 19:50, 5 June 2024 (UTC)
 No, there are not precisely two definitions of the natural numbers. There are multiple ways, broadly descrived as synthetic (axiomatic) and constructive. The conventional synthetic definition of using the Peano postulates} is the same whether 0based or 1based, differing only in the choice of symbol for the initial element. However, there are other synthetic approaches using axion schemes from, e.g., John von Neumann, Raphael M. Robinson.
 For constructive definitions, the definitions typically start by defining an initial element and a succesor function, then applying induction. I.e., .  Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:32, 5 June 2024 (UTC)
 I think you are confusing definitions with formalization. The definitions I am talking about are "0,1,..." and "1,2,...". These are short and concise and about as good as definitions can get. In terms of formalizing, the "..." is what causes the issue and the ways of dealing with infinity are relatively tangential to defining the natural numbers. Per WP:TECHNICAL the current article doesn't start out by defining the natural numbers using the Peano postulates or whatever, it gives the straightforward definitions. Mathnerd314159 (talk) 18:28, 5 June 2024 (UTC)
 @D.Lazard With the color stuff, there is a good refutation of your argument. Consider applying your argument black and white. "The fact that the definitions are very similar" can be seen on their wiki pages: "one passes from one definition to the other" by changing "the complete absence of visible light" into "the full presence of all wavelengths of visible light". So, just like the presence or absence of 0 is a minor irrelevant detail, "it is much clearer" to have a single article Grey that states clearly that there are two extremist definitions of the color. The flaws in this argument should be evident. Mathnerd314159 (talk) 17:46, 5 June 2024 (UTC)
 No, that is not what the Banach–Tarski theorem says. It's the pieces that are isomorphic, not the balls themselves.  Shmuel (Seymour J.) Metz Username:Chatul (talk) 16:32, 5 June 2024 (UTC)
 BanachTarski proves the existing of a bijective mapping which maps every point of the single sphere to a corresponding point of one of the two spheres. And then it is "natural" in the sense that it consists only of rotations and translations and preserves almost all of the local structure.
 It is sort of like what I am proposing here, dividing up an article into two distinct concepts, primarily using cutandpaste, in a way that preserves the flow of the text. Mathnerd314159 (talk) 18:00, 5 June 2024 (UTC)
 Natural numbers form a single concept, with two slightly different definitions. So WP:NOTDICT does not apply. The fact that the definitions are very similar can be seen on Peano axioms, the standard way to formally define the natural numbers: one passes form one definition to the other by changing the axiom "0 is a natural number" into "1 is a natural number" (and vice versa), and keeping the other axioms unchanged. Moreover, many mathematical texts do not state clearly which definition they use, generally because this does not matter. So a dab page (that you suggest implicitely) would be confusing, since most readers would not know which link to choose. So, it is much clearer to have a single article that states clearly that there are two variants in the definition. D.Lazard (talk) 21:22, 4 June 2024 (UTC)
 You say they are "the same", but then they are written , , clearly different. Under the Banach–Tarski paradox one ball is isomorphic to two balls, are you going to make 2 redirect to 1 because "there is no mathematical difference"? Mathnerd314159 (talk) 20:32, 4 June 2024 (UTC)
 Oppose This is just one concept, with two slightly varying definitions. We can (and do) easily cover the two definitions in one article. There's no need for two articles here any more than there is a need to split Prime number into three articles to cover other historical definitions that included 1 as a prime, or did not include 2 as a prime. Meters (talk) 02:48, 5 June 2024 (UTC)
 Strongly Oppose Call for an end to a discussion that seems to have reached a conclusion. Rick Norwood (talk) 10:07, 5 June 2024 (UTC)
 There is WP:STRONG which points out "[The word "strong"] does nothing to actually strengthen your position or influence what consensus comes out of the discussion, especially if it is all you have to say. A clear and logical statement of your position, with or without a bolded comment prefacing it, is what makes your argument strong."
 As far as discussion length, typically a week is the minimum, but I was thinking more like two months based on the previous thread #Is 0 a natural number?. I guess a week is fine if no other supporters come forward. Mathnerd314159 (talk) 18:05, 5 June 2024 (UTC)
 Strong Oppose with a side of touch grass. This proposal has no foundation in guidelines, policy, or common sense. There are not two topics here; there is one topic with two alternate conventions. There is no dividing line to split the material along. A split would produce no clarity and make both maintenance and improvements more difficult. A split would violate NPOV by making the choice of convention seem more important than it is. I tried a SNOW close, because there were six editors advocating against the proposal in this section and two more in the previous, indicating that there was the proverbial snowball's chance of a consensus actually forming in favor of it. But I was reverted by the proposer of and only advocate for the split. XOR'easter (talk) 23:18, 5 June 2024 (UTC)
 Here is the proposed split: https://ibb.co/3hgLRTS. Yellow to stay here, red to go to nonnegative integer, blue to go to positive integer. Pretty disingenuous to say there is no dividing line when it is so obvious. And of course positive integer is a little short but I am sure it can be expanded, e.g. with a discussion of how positive integers are learned in early childhood. Mathnerd314159 (talk) 23:38, 5 June 2024 (UTC)
 Thank you for illustrating how little sense the split would make, and in color at that. The yellow includes discussion of both conventions. Ergo, both conventions belong on the same page, the way it already is done. The long stretch of red also indicates that the bulk of the "Properties" section would have to be duplicated between the "positive integer" and "nonnegative integer" pages. You're advocating a division without foundation. XOR'easter (talk) 23:56, 5 June 2024 (UTC)
 Here, you said
it has not even been 24 hours
. Yes, it has. You made your split proposal at 05:23 the morning of the 4th (UTC). One day after that, the proposal had already accrued 5 oppose !votes in this section and two more oppositions just above. XOR'easter (talk) 00:24, 6 June 2024 (UTC)
 Here is the proposed split: https://ibb.co/3hgLRTS. Yellow to stay here, red to go to nonnegative integer, blue to go to positive integer. Pretty disingenuous to say there is no dividing line when it is so obvious. And of course positive integer is a little short but I am sure it can be expanded, e.g. with a discussion of how positive integers are learned in early childhood. Mathnerd314159 (talk) 23:38, 5 June 2024 (UTC)
 Oppose for the reasons already expressed atlength above. Walsh90210 (talk) 00:30, 6 June 2024 (UTC)
 Oppose Book authors choose one or the other convention, but they don't have two chapters for natural and "more natural" numbers  why? because it's the same concept. We'd practically have a POV WP:CFORK with two articles repeating all the same things in two slightly different ways. That would be confusing for our readers and too messy to sync and maintain. Ponor (talk) 03:36, 6 June 2024 (UTC)
Shouldn't this article spend more time on uses of the natural numbers?
[edit]It seems to me that it would be valuable to have a section or sections discussing the way natural numbers are used, rather than only describing their properties.
We should probably have some discussion of how natural numbers are used as a basis for counting, sorting, comparing sizes of finite collections, etc. Are there some particularly important results that are reasonably accessible that could give readers a flavor of what natural numbers are for and what they are about? We might e.g. talk about the dimension of a geometric space, the order of a group, the rank of a matrix, ... We could briefly discuss the cardinality of the natural numbers and how it relates to the cardinalities of other sets (notably, the rational numbers being enumerable but not the real numbers).
Beyond natural numbers per se, they can also be used as a "building block" to construct other kinds of numbers: for instance the integers can be defined as equivalence classes of pairs of natural numbers, where two pairs are considered equivalent if the pairs are the same distance apart; the rational numbers can be defined as equivalence classes of pairs constructed from an integer numerator and a positive integer denominator; the integers modulo n can be taken to be equivalence classes of natural numbers (or integers) where numbers n apart are considered equivalent; the Eisenstein integers can be defined as equivalence classes of triples of natural numbers where triples are considered equivalent if adding the same number to each element in one triple yields the other triple. Etc. etc.
–jacobolus (t) 02:40, 6 June 2024 (UTC)
 Absolutely: add whatever you have in your sources for natural numbers. (but don't go too deep  in this article) Ponor (talk) 03:42, 6 June 2024 (UTC)
Article seems to confuse two different concepts
[edit]I don't think it makes sense to discuss both the nonnegative 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 nonnegative 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)
 <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)
 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 nonnegative 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)
 "Can be said" is not the same as "want to say". Sure, there's a fair amount you could potentially say about offbyone 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)
 The situation here is kind of similar to what you describe with the offbyone, 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 welldefined, 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)
 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)
 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 applestooranges 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)
 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)
 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 nonnegative 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)
 ^ 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