Talk:Natural number

Credit for axioms: Peano versus Dedekind

The so-called Peano axioms should be credited to Richard Dedekind, as was done by Giuseppe Peano himself, and is currently done in the Wikipedia article Peano axioms. Dedekind published the axioms in 1888, before Peano. I've done a quick fix, which should be filled out with proper referencing, but there is no controversy about the basic facts. Crediting the axioms to Peano is simply a mistake, not a considered dissent, however prevalent this practice may be.

The practice of axiomatizing the natural numbers (as opposed to the specific Peano axioms) was introduced by Charles Sanders Peirce in 1881; again, this is credited to Peirce, not Peano or Dedekind, in the article Peano axioms.

—Syrenka V (talk) 05:11, 31 March 2018 (UTC)

Notation section

Right now, the various notations for the natural numbers with or without 0 are scattered in several places: in the second paragraph, at the end of "Modern definitions", and in "Notation". Should these be consolidated? Also, perhaps it is worth adding the notations $\mathbb {Z} _{\geq 0}$ and $\mathbb {Z} _{>0}$ to the list; in my experience, these are more common than some of the other alternatives listed. Ebony Jackson (talk) 02:00, 16 January 2021 (UTC)

Varions notations for the set of natural numbers should not be in the head. A sentence about the existence of such a set should be added before its notation. CBerlioz (talk) 09:37, 19 September 2021 (UTC)

Whole numbers = integers?

The article says "in other writings, the term (whole numbers) is used instead for the integers (including negative integers)", but it gives only one citation, and even that citation is defining "integer" to mean "whole number", not vice versa. I suspect that including negative integers in the whole numbers is nonstandard, a convention used only by a fringe. Does anyone have references to suggest otherwise? Ebony Jackson (talk) 04:03, 25 January 2021 (UTC)

At the very least "that term" is unnecessarily ambiguous. I suspect it refers to "whole numbers". The whole sentence could use some cleanup. There should be a separate reference for each of the uses. BFG (talk) 14:58, 25 January 2021 (UTC)

(Heavy sigh) This again. I really wish we could just ban the term "whole number" and not let it appear anywhere in Wikipedia. That would fix the issue good! And nothing would be lost in terms of modern serious mathematics, because the term "whole number" is essentially unused in that context.

Unfortunately it is used in other contexts, and with inconsistent meanings. The one I learned in California K-12 education was that the natural numbers exclude zero and the whole numbers include it. Since the more useful definition of "natural number" includes zero, that one is superseded.

However it seems that other writers use it to mean natural numbers excluding 0, or integers (including negative). See the archives of talk:Whole number to find tedious discussion on this point, possibly with refs. --Trovatore (talk) 20:43, 25 January 2021 (UTC)

Update: Just looked through that talk page and its one archive and nothing jumped out at me as useful. Maybe the archives of this talk page, or of talk:integer? I know it was somewhere. --Trovatore (talk) 20:45, 25 January 2021 (UTC)

OK, this may be what I was remembering: talk:natural number/Archive 2#Implementation of whole number to redirect here. It seems that Maproom particularly was arguing for the existence of the "whole number==integer" meaning; if he/she is still around, perhaps we will hear more.

I see that Maproom presented two references for this usage, neither of which is really ideal; one was a dictionary, and the other was Mathworld. Honestly I would be inclined to discount Mathworld almost completely on issues of usage. Mathworld is a reasonable site for mathematics, but really an appallingly bad one for nomenclature. And both Mathworld and dictionaries are tertiary sources, which are not as good as secondary sources. That said, it seems to me that there's very little harm in saying that "whole number" can sometimes have the sense of "integer", given that (1) it seems to be true, for some value of "sometimes", and (2) the term "whole number" isn't used much in the serious mathematical corpus, so it doesn't matter too much if people allow it meanings that it rarely has. --Trovatore (talk) 06:18, 26 January 2021 (UTC)

If I once had an opinion on this, I've forgotten it. I would recommend writing "integer", "non-negative integer" or "positive integer", and avoiding "natural number" and "whole number" as ambiguous. If the article is to mention the ambiguous nature of terms, it should supply references – they can't be hard to find. Maproom (talk) 08:17, 26 January 2021 (UTC)

Just to give some perspective on this: The German word for 'integer' is 'Ganze Zahl', which translates to 'whole number', so I could see how this could cause some confusion, at least for a non-native English speaker like me. :) Phonous (talk) 23:41, 4 April 2021 (UTC)

In French also, the word for 'integer' is nombre entier often abbreviated as entier, whose literal translation are 'whole number' and 'whole'. The term for 'natural number' is nombre naturel or entier naturel; the first term translates literally to the English, and the second one could be translated 'natural whole' or 'natural integer'. The term entier relatif ('relative integer') is also used for "integer", for emphasizing that negative integers are included. All of this suggests that the term "whole number" results from the influence of other European languages, and this may explain its different uses. D.Lazard (talk) 08:03, 5 April 2021 (UTC)

The use of the phrase "whole number" is just one more example of the disconnect between K-12 usage and professional usage in the United States. I do not know if this is reflected in other English speaking countries. I have never heard a STEM professional use the phrase "whole number". Professionals either use natural number or integer. But K-12 teachers, taught the phrase "whole number" themselves, insist on teaching it to their students. If Wikipedia mentions it at all, it should say, "Whole number" is a phrase used in K-12 education with inconsistent meanings, some of which include the meaning "Natural number", "non-negative integer", and "integer". Rick Norwood (talk) 11:17, 5 April 2021 (UTC)

Just weighing in I corrected the page Whole number to mention that it is a colloquial term. D.Lazard points out the word in German is 'Ganze Zahl' which roughly translates to whole number. The use in German should carry some weight, as both $\mathbb {N}$(Natürliche Zahl) and $(\mathbb {Z} )$(Ganze Zahl) takes their letter from German. This nomenclature is reflected in all the Scandinavian languages, but in French the names translates to literally natural integers and relative integers. I would claim that the use of whole number thus should mean the integers $(\mathbb {Z} )$. But that is just my personal opinion, I think it's better to only talk of it as a colloquial term. BFG (talk) 12:46, 5 April 2021 (UTC)

I did a google ngram search comparing the frequency of "natural numbers" versus "whole numbers" and it appears that up to about 1960 "whole" was rather more frequent, and "natural" started somewhat dominating since only about 2010. So whatever meaning is attached to the terms, above conclusions do not seem warranted.−Woodstone (talk) 13:28, 5 April 2021 (UTC)

I don't think that's a valid comparison. This is a mathematics article and should give priority to the usage in serious mathematics. A start might be to limit the search to Google Scholar; not sure if that's feasible with ngrams. --Trovatore (talk) 17:09, 5 April 2021 (UTC)

Whole numbers are taught in K-12, integers in college. Far more people go to K-12 than go to college. That explains the frequency of use. But, as Trovatore points out, Wikipedia articles on mathematics report the professional use, even when it goes against what is still taught in many K-12 classes. Rick Norwood (talk) 11:24, 6 April 2021 (UTC)

In my entire academic life, I don't recall once encountering a person or a textbook that equated the "whole numbers" with the integers but many times where it was defined as either the positive or non-negative integers. I concur with the tagger this is a very dubious, non-standard claim. Jason Quinn (talk) 12:53, 8 May 2021 (UTC)

Formal and intuitive définitions.

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 set-theoric 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

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 set-theorical 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 second-order-logic 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

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

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 Set-theoretic 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, North-Holland 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 non-specialists, 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 § Set-theoretic 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

Considering the natural numbers $\mathbb {N}$ to be cardinals and ordinals only makes sense if $\mathbb {N}$ includes zero, since the null set $\emptyset$ 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 secondary-school 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 primary-school textbooks, that's a legacy of the so-called New Math, a well-intentioned 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, do not use the term null set for a set of measure 0 and various online sources, e.g., Britanica, MathWorld, list null set as synonymous with empty.

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. K-12 texts are typically not written by SMEs. -- Shmuel (Seymour J.) Metz Username:Chatul (talk) 13:41, 18 March 2024 (UTC)

References

Finite

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?

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 recently-published 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" 1-based 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_pre-algebra_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 1-based 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)

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 0-based 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 1-based 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 1-supporting sources I gave. Looking at recently-published books, it seems split between university-level (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 elementary-level sources as "trash" and say that the university-level 0-based 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 non-negative 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 1-based 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 0-based or 1-based 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 0-based has become the norm among more recent books. Mathnerd314159 (talk) 20:47, 19 February 2024 (UTC)

There are two approaches to $\mathbb {N}$, 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 1-biased phrasing to something giving 0-based 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)

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)

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 high-school level) to exclude 0 from the natural numbers, and for upper-division college-level 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 fact-checking 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 theory-adjacent 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 non-negative 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 number-theory texts start with 1 and these set-theory 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 non-negotiable, 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 university-level 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)

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 1-biased 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)