Talk:Natural number

2985:) 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 2981:

inverse elements without including ℤ, the negative integers. I hope you'll excuse me, it's been a while since I did my degree (1993-6, 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 10ᵗʰ-11ᵗʰ Century; qv

2823:). 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 one-one 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? 2909:

begins from 1.(also if you use p-adic 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 p-adic 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.

3663:, 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. 449:, 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. 348: 330: 256: 2588:, 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 Knowledge's POV. 1889:. 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 246: 225: 192: 3073:(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). 3200:

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 Knowledge, 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’.

2581:, 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. 3827:"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 non-negative integer definition. So a disambiguation page with two links is the most structured representation of the meaning of the natural numbers. 3924:

and arguments and then wikilawyering about the straw man version is not helping convince anyone. Instead of making increasingly implausible-to-happen 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. –

3016:. There is no debate about what the "positive integers" or "non-negative 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 non-negative 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 3730:." I think it is clear that the positive integers and the non-negative integers are independent, discrete subjects and meet GNG. I also think Knowledge would be clearer if it consistently used positive integer / non-negative 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. 3909:- 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. 4283:. 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 152: 3273:(currently both redirects to here). The content will be split among them according to whether it uses the 0-based or 1-based 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. 2147:, 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. 2869:(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. 183: 2696:, 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 2990:

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 Knowledge users looking for a definition.

2472:"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. 4145:. "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 4552:, it is 3 paragraphs. There are definitely 3 paragraphs of information each about the positive integers and the non-negative integers. And there are definitely a ton of reliable sources, so they are both notable. To me it is obvious they should be separate articles. But apparently 8 people disagree. 8 vs. 2 now, maybe it is time for another split proposal. 😄 1137: 1087: 3181:

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).

3302:. 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. -- 1546:, 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. -- 4428:

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).

4569:

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,

4480:

I don't think it makes sense to discuss both the non-negative integers and the positive integers in the same article, when integers gets its own article. The difference may only be a single element (namely, zero)—but this single member is of great importance, e.g. when discussing factorization, being

4431:

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

3525:

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

2934:

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

2865:

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).

2841:

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 '66-Jul '67 – and one in the USA – Aug '67-Jul '68; academic years: my father partook in exchange lectureships). I do

2543:

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

2519:

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

2471:

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):

2339:

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

2259:

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 Knowledge, 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

2161:

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,

1051:

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

817:

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

482:

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

4496:

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

4427:

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

3923:

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

3695:

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

3199:

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

3076:

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

2989:

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

2938:

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

2133:

on Knowledge 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

1000:

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

543:

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

507:

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

4621:

Mathnerd, you're wasting your time here. You've repeated the same couple of points now ad nauseam, while throwing in a mishmash of irrelevant apples-to-oranges comparisons, non sequiturs, and straw men, but it's not convincing anyone. If 8 people trying to explain why this seems like a bad idea was

4592:

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

4500:

This article is not a discussion of the word, or the term, it is a discussion of a specific mathematical concept. The fact that, historically, the non-negative integers (or integers, or positive integers) has at times been called the "whole numbers" and/or "natural numbers" is a fact to list in the

3799:

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.

3699:

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

3437:

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

3098:

The Knowledge 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

3080:

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 tom-toms, 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

3057:) 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 2886:

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

2602:

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

1004:

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,

478:

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

3430:

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

2504:

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"

2180:

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

418:

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

3744:

The "single concept" here is an ordered, countably-infinite 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

2908:

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

2669:

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

2617:

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

1841:

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

1242:

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

4332:

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

2361:

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

2026:

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

4588:

The situation here is kind of similar to what you describe with the off-by-one, most of the article is about nonnegative integers and then there is some stuff about positive integers unnaturally mixed in. It is true no one studies "the natural numbers with zero" and "the natural numbers without

3180:

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

3115:

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

2673:

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.

2980:

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

2942:

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.

2562:

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

2195:

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

899:

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,

633:

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. »

4106:

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

649:

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

2997:

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 light-hearted, 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

3438:

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. –

856:

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.

4252:

which points out " 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

3361:. 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. 3875:

In your own textbook, please pick whichever one you prefer and stick to it. In Knowledge, we should briefly mention that conventions differ (if only to forestall time-wasting complaints from pedants), and then move on to discuss the actual subject: the use and structure of these tools.

2704:: "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 4593:

past the basic Peano axioms, nothing translates without major changes or adding ugly conditions like "≠0" - for example, exponentiation on positive integers is well-defined, but 0^0 is not. If it really was completely equivalent there would not be a debate, there would be a theorem.

2432:, 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" 4183:

Banach-Tarski 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

2281:

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

1530:

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

650:

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.

3872:. 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 well-defined list that you can use as a basis for comparison, but where precisely it starts is a triviality. 1701:

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

3707:"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 1035:

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.

2993:

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

2065:

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

3450:"The names 'non-negative 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 2304:

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. --

733:

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). »

3077:

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).

2866:

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.

441:

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

1880:

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

872:

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

2632:

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.

2138:

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

2376:

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.

2947:

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 0-based and which are 1-based, so splitting it would not be hard. Thoughts, anyone?

570:. 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 3830:

And it is not "we can quibble", it is "we HAVE quibbled", for several centuries (and 20 years on-wiki), over which of these two definitions is appropriate. The other things you mention such as rational numbers or arithmetic have not been an issue.

2015:

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

2106: 4287:

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

2344:

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).

2935:

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.

1864:

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?

530:. 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. 2538:

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

933:

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

4318:. 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. 2263:

This is not a question Knowledge 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.

2022:

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.

837:(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 3156:

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

3891:"The important part is to have a particular well-defined list" - I agree, but since the natural numbers are not well-defined the only way to get a well-defined list is to avoid the term entirely and use more specific terms. 1333:

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.

883:

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.

3951:

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.

2027:

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.

1227:. 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. 4570:

the study of which has been the principal preoccupation of the entire professional lifetimes of many many brilliant people. None of those people divide that study into the structure with or without zero. They

4488:

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.

1270:

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

4530:

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

2413:

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.

2108: 2135: 1882: 1819:

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

1535:; 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. 4212:

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

2196:

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.

2260:

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.

4354:. 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. 3405:. Certainly this would argue that "non-negative 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 2142:

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

1978:

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."

3745:

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

2756:

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,

2240:, 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 4444:

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.

2798:

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?”.

4626:

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. –

4566:"Can be said" is not the same as "want to say". Sure, there's a fair amount you could potentially say about off-by-one errors, but it doesn't live naturally in an article about the natural numbers. 4497:

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.

1274:

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

793:

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

1994: 479:

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.

3966:

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.

3800:

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. –

2548:

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.

3232:

It is clear that there is consensus against this split proposal, and that there is no realistic hope of that changing. Therefore it is hard to see any benefit of continuing to discuss it. --

3888:"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. 4589:

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.

4090: 3409:

would not be a disambiguation. If we include "natural number" in the non-negative integer page then Google will most likely pick up on it and give it as the top result for "natural number".

312: 1349:

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

2008: 1886: 619:

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.

4392:

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

985:

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.

4730: 3704:) 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. 2100: 4022:} is the same whether 0-based or 1-based, differing only in the choice of symbol for the initial element. However, there are other synthetic approaches using axion schemes from, e.g., 2665:

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

2102: 3053:

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 Knowledge's entry on

2122:

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,...}.

3403: 2099:

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

3284:

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 non-negative integers.

2603:

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.

3454:. 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. 3320:

Precise language and definitions can be a matter of life and death. Not sure there are any specific examples of zero vs. one-based, as it typically is caught early in testing, but

3203:

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.

3084:

Please accept my apologies for going so far off-topic, it's just that I do enjoy these discussions. This is a ‘talk’ page, after all, so I did not think it entirely inappropriate.

2816:

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.

2618:

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, ..."?

3627: 3599: 1052:

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 ?

2104: 483:

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.

1620:

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.

2842:

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 🙏🏼.

1149: 4187:

It is sort of like what I am proposing here, dividing up an article into two distinct concepts, primarily using cut-and-paste, in a way that preserves the flow of the text.

4014: 3081:

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.

2726:

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. --

2238: 2181:

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.

1419: 1397: 604:

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.

3905:"In Knowledge, 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 143: 1439: 4548:

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

3868: 2939:

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.

1775:

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

4720: 1005:

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.

3992:

definitions of the natural numbers. There are multiple ways, broadly descrived as synthetic (axiomatic) and constructive. The conventional synthetic definition of

3768: 880:

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.

589:

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 … ».

4735: 4424:

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.

3792: 1634:

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. --

3138:

a debate about what the "positive integers" are; some authors use it to mean non-negative and use the term "strictly positive" when the want to exclude zero. --

3715:

to turn it into a broad-concept page that actually explained when it was used as integer vs. positive integer vs. non-negative integer was opposed by Trovatore)

3338:

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. --

3265:

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

4481:

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.

2887:

and "the natural numbers" the other way. But the main point is that, of all the ways of deciding this question, deciding it on Knowledge is probably the worst.

4755: 1827:

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.

388: 4396:

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.

3020:. 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. 3845:

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

1224: 833:"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 4512:. Alternatively, appropriate sources could be added to show why these two sets are, in fact, the same mathematical concept deserving of a single article. 3726:, "Merging should be avoided if: ... The topics are discrete subjects warranting their own articles, with each meeting the General Notability Guidelines, 3402:

says 48% of traffic is from search. In Google Trends, it is maybe 90-60-20-1 for whole number / natural number / positive integer / non-negative integer.

2998:

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 🙏🏼.

489:

In formalized mathematics, the natural numbers are both: - the terms of axiomatic arithmetic. - a construction of set theory satisfying the Peano axioms.

4745: 2545: 915:

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

302: 79: 4111:

the current article doesn't start out by defining the natural numbers using the Peano postulates or whatever, it gives the straightforward definitions.

2244:. 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. -- 1189:“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). 4715: 2144: 4432:

numbers can be defined as equivalence classes of pairs constructed from an integer numerator and a positive integer denominator; the integers modulo

4033:

For constructive definitions, the definitions typically start by defining an initial element and a succesor function, then applying induction. I.e.,

3495:

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,

394: 1282:

elements". So, all your advanced considerations, could be replaced in this article by "with von Neumann's definition of the natural numbers, a set

4725: 4493: 3526:

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.

3824:. 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. 2486:

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.

278: 44: 4333:

would have to be duplicated between the "positive integer" and "non-negative integer" pages. You're advocating a division without foundation.

1542:

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

4760: 4740: 2924: 2449:

OK thanks. So not ancient, but still nearly a century before Peano was born, then. Might be worth working this into the article somehow. --

85: 4508:, which would discuss related subsets and how that affects their mathematical properties; or the relevant portions of this article moved to 2057:

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

1170: 900:

instead of the end of the section modern definitions: the existence of the set of natural numbers is guaranteed by the axiom of infinity ?

2744:, 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 2524:

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. --

4688: 1998: 1198:

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.

1020:

Would you agree with the formulation: “Naïve set theory admits that the natural numbers form a set, to which mathematicians refer …” ?

4454: 3434:

The names "non-negative integer" and "positive integer" needlessly exclude the other set, which should be discussed in the same place.

2177: 1145: 1129: 364: 4680:

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

4501:

relevant article whatever its name may end up being; this does not make an excuse to combine the two concepts into the same article.

3948:

is not consensus, or at least it is the weakest form and the presence of a disagreeing editor such as myself disrupts that consensus.

1511:

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

4750: 3936:

The situation was not "the overwhelming consensus of editors", it was established by user Dissident in 2004. The choice to redirect

1972:

Talk:Natural_number/Archive_4#Problem_with_notations_:_the_notations_of_this_page_do_not_respect_international_standard_iso_notation

1676: 269: 230: 4256:

As far as discussion length, typically a week is the minimum, but I was thinking more like two months based on the previous thread

196: 4710: 1262:

given in this article is based on cardinals as well as on ordinals, since only finite sets are considered here, and the number

99: 30: 2110:

all use 1, these span from 1800's to 1966. There are several references that attribute the 0-based numbering to Bourbaki 1968

104: 20: 3099:

or physical properties of objects. Beyond that, as you point out, the word "black" applies to a wide range of perceptions. –

419:

number means formally: - a primitive notion of axiomatic arithmetic. - a set-theoric construction satisfying Peano axioms.

4654: 4363: 4342: 4301: 3975: 3260:

Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

2335:

of definitions that are most commonly used in very different contexts. By my count 34 of the past 50 edits (23 of them by @

4415:

Subsequent comments should be made on the appropriate discussion page. No further edits should be made to this discussion.

1971: 1966: 1961: 1956: 1951: 1946: 1941: 1936: 1931: 1926: 1921: 1915: 1910: 355: 335: 139: 135: 131: 127: 74: 4149:

that states clearly that there are two extremist definitions of the color. The flaws in this argument should be evident.

1069:“Natural numbers form a set. Many other number sets are built by successively extending the set of natural numbers: ….” 205: 4574:

for definiteness, but recognize that everything they say would translate with minor changes to the other convention. --

4217:

into three articles to cover other historical definitions that included 1 as a prime, or did not include 2 as a prime.

3280:. Under criteria, it lists that the proper resolution of "the same title for different things (homographs)" is to have 3246: 65: 938:

could be taken as a definition, but while this might fit in the body somewhere, it's not appropriate for the lead. --

559:

This is definitively not for the beginning of the lead. However, one could add the following at the end of the lead:

2705: 1842:

this is the case then it probably shouldn't link to cardinal/ordinal number as those are not the intended meanings.

2111: 838: 151: 122: 4166: 4036: 3630: 4460:

Absolutely: add whatever you have in your sources for natural numbers. (but don't go too deep - in this article)

3682: 3186: 2828: 2477: 2563:

numbers". In fact, the choice of including 0 or not depends mainly of convenience in each mathematical context.

749:

I agree with Trovatore's deletion and his assertion that this kind of sentence is not appropriate for the lead.

162: 4598: 4557: 4323: 4265: 4192: 4154: 4116: 3957: 3914: 3836: 3796:), but the basic purpose and idea is the same in any event, and the exact starting point is largely irrelevant. 3735: 3646: 3531: 3504: 3459: 3414: 3329: 3289: 3025: 2967: 2953: 2920: 2874: 2713: 2623: 2593: 2553: 2510: 2491: 2440: 2350: 2201: 2167: 2152: 1984: 1898: 1847: 1204:

Definition: a cardinal n is a natural number if and only if there exists a finite set x such that n = Card(x)”

1165: 4607:

OK, you're again descending into quibbles that make it hard for me to believe you're taking this seriously. --

463:

No, as the sentence refers to the most technical content of the article, it should be at the end of the lead.

4692: 3037:– as an off-topic aside, black is most certainly a color by any plausible scientific definition of "color". – 955:

It’s quite elementary to assess or prove the existence of a set (for example Paul Halmos, Naïve Set Theory).

2986: 2944: 2500:

As far as audience, you cannot really assume anything about the reader's level. There is some discussion in

1890: 109: 2081:

Mathnerd314159's suggestion: "the natural numbers are the numbers 0, 1, 2, 3, etc., possibly excluding 0."

4239: 3817: 2916: 2895: 2367: 2332: 2269: 2035: 1563:

I'm not familiar with contemorary texts, but the ones I'm familiar with, e.g., Rudin, do not use the term

3551:

The two systems of natural numbers are naturally isomorphic and are often described in terms of the same

778:

Alternative places are the head of Modern definitions section or the head of Formal definitions section.

4650: 4359: 4338: 4297: 3971: 3945: 3895: 1001:

the natural numbers form a set or not. It is pedantry to introduce here advanced concepts of set theory.

797:. Please, stop triyng to push your point of view against a clear consensus (three editors against you). 767: 696: 672: 534: 486:

My new proposal is: « In informal mathematics, the natural numbers are those … in a mathematical sense.

211: 4549: 3821: 347: 329: 255: 3604: 3576: 2670:

starts with 1 in number theory and 0 in set theory, is the kind of original research we can't do here.

1152:

until a consensus is reached, and readers of this page are welcome to contribute to the discussion. –

561:

The definition of natural numbers has been formalized in several essentially equivalent ways, through

4380: 4311: 3941: 3678: 3270: 3196: 3182: 3116:

colour 😉? Anyway, thanks again. I'm always flattered when someone takes the time to reply to me ☺️.

2912: 2838: 2824: 2819:

My understanding is that zero was introduced historically as a place holder (see, e.g., section 1 of

2795: 2608: 2604: 2578: 2473: 2336: 1692: 688: 182: 4630: 4612: 4594: 4579: 4553: 4539: 4451: 4441: 4319: 4261: 4188: 4150: 4137:

With the color stuff, there is a good refutation of your argument. Consider applying your argument

4112: 4108: 4027: 3953: 3928: 3910: 3880: 3832: 3804: 3731: 3642: 3527: 3517: 3500: 3483: 3455: 3442: 3410: 3343: 3325: 3307: 3285: 3237: 3103: 3041: 3021: 2977: 2963: 2949: 2870: 2731: 2709: 2679: 2638: 2619: 2589: 2549: 2529: 2506: 2501: 2487: 2454: 2436: 2418: 2382: 2346: 2341: 2328: 2293: 2197: 2163: 2148: 1980: 1894: 1843: 1551: 1335: 1244: 1205: 1192:

Axiom of cardinality (P.Suppes): The sets A and B are equipollent if and only if Card(A) = Card(B)

1153: 1114: 1070: 1053: 1021: 986: 975: 956: 943: 901: 858: 819: 779: 735: 723: 635: 620: 590: 545: 493: 438: 423: 55: 437:

The mathematical definition of natural numbers does not belong to the first sentence. Please read

363:

on Knowledge. If you would like to participate, please visit the project page, where you can join

277:

on Knowledge. If you would like to participate, please visit the project page, where you can join

4623: 4517: 3997: 3906: 3723: 3668: 3366: 2765: 2568: 2433: 2392: 2221: 2130: 2086: 1832: 1520: 1471: 1402: 1380: 1362: 1339: 1321: 1307: 1248: 1232: 1209: 1118: 1103: 1074: 1057: 1041: 1025: 1010: 990: 960: 924: 905: 889: 878:

Peano arithmetic is equiconsistent with ZFC with the axiom of infinity replaced by its negation.

862: 846: 823: 802: 783: 754: 739: 655: 639: 624: 609: 594: 579: 549: 517: 497: 468: 454: 427: 261: 167: 70: 2748:, since it is a primary source, and we do not have any secondary source discussing it. So, both 245: 224: 1424: 4638:

The fact that a few things depend upon one's choice of convention doesn't mean that there are

4235: 3712: 3656: 3496: 3321: 3277: 3208: 3165: 3121: 3089: 3062: 3013: 3003: 2891: 2847: 2807: 2363: 2265: 2096: 2031: 1673: 1625: 51: 4327: 3850: 3711:. If you are going to wage a crusade against confusing dab pages you should start there. (My 2708:)—instead believe that defining the natural numbers as starting from 1 is more appropriate." 2435:. The concept of "natural number" requires the concept of "unnatural numbers" as a contrast. 718:. I think most readers will understand the former and may not have even heard of the latter.— 4646: 4355: 4334: 4315: 4293: 4249: 4222: 4174: 4097: 4023: 4019: 3967: 3937: 3813: 3560: 3552: 3451: 3427: 3324:

is an example of the general gravity of the situation. It is not "silly" or "a triviality".

3266: 3160:

We are, kind of, back to language and that's tricky at best, as anyone bilingual will know…

3143: 3070: 2656: 2400: 2310: 2249: 2119: 1870: 1809: 1748: 1724: 1639: 1605: 1499: 1491: 1450: 1186:

The construction based on cardinals could be added, for example with the following wording:

764: 692: 669: 531: 164: 2429: 2113: 4465: 4401: 4393: 4376: 3899: 3719: 3396: 3066: 2697: 2540: 2324: 2058: 1785: 562: 2740:

Saying "0 is sometimes included and sometimes excluded" is clearly conform to the policy

2323:

that if there are differing viewpoints they should all be mentioned. The issue is giving

4534:

As for "whole number", that's a term that is not much in use in research mathematics. --

4627: 4608: 4575: 4535: 4448: 3925: 3877: 3801: 3750: 3514: 3513:

It seems to me like you are mad that people disagree with you and are flailing a bit. –

3480: 3439: 3399: 3392: 3339: 3303: 3233: 3112: 3100: 3058: 3050: 3038: 2757: 2749: 2741: 2727: 2689: 2675: 2634: 2585: 2525: 2450: 2415: 2378: 2320: 2289: 2186: 2046: 1801: 1791: 1547: 1463: 1354: 1350: 1259: 1220: 971: 939: 794: 719: 571: 24: 4436:

can be taken to be equivalence classes of natural numbers (or integers) where numbers

3777: 2505:

and the elementary school teacher can say "... sorry, that's what our textbook says."

4704: 4513: 4134: 3664: 3362: 2761: 2753: 2564: 2082: 1828: 1770: 1516: 1467: 1442: 1358: 1317: 1303: 1228: 1099: 1037: 1006: 920: 885: 842: 798: 750: 683:

Is it possible to paraphrase "Intuitively" as other words ? Because it seems to mean

665: 651: 605: 575: 527: 513: 464: 450: 3894:"In your own textbook, please pick whichever one you prefer and stick to it." - I'm 3065:(pigments) colouring – the primary colours of the former, eg in your TV screen, are 2162:

who define the natural numbers to encompass the positive integers (1, 2, 3, etc.)."

1567:

for a set of measure 0 and various online sources, e.g., Britanica, MathWorld, list

4509: 4485: 4214: 3708: 3660: 3655:

Natural numbers form a single concept, with two slightly different definitions. So

3406: 3204: 3161: 3117: 3085: 2999: 2843: 2803: 2745: 2126: 2071: 2012: 1962:

Talk:Natural_number/Archive_4#Distinction_between_whole_numbers_and_natural_numbers

1927:

Talk:Natural_number/Archive_2#Do_number_theorists_start_the_natural_numbers_with_1?

1621: 916: 684: 3555:. From a mathematical perspective it makes no sense to treat them differently. -- 668:. As I've said above, there is no need to mention formal definitions in the lede. 4218: 4170: 4093: 3701: 3556: 3388: 3153: 3139: 3012:

Splitting would put different definitions on different articles, as required by

2652: 2396: 2306: 2245: 1866: 1805: 1669: 1635: 1601: 1495: 1487: 1446: 274: 2820: 1441:

has cardinality 0 and order type 0. Should the article discuss that or is that

876:

Moreover, the sentence that I have reverted is contradictory with the article:

4461: 4397: 3659:

does not apply. The fact that the definitions are very similar can be seen on

1714: 1113:

Construction based on cardinals could be added in section Formal definitions.

1095: 566: 251: 1738: 1066:

Another (non exclusive) solution could be at the beginning of 3rd paragraph:

4419: 3379:

In popularity on Knowledge, "natural number" is about the same pageviews as

2241: 2182: 2070:?), since this simplifies many notations (for example, the zeroth term of a 1743: 1719: 1696: 1512: 1291: 4676:

This sort of categorical statement is always risky; I imagine you can find

1195:

The definition of a finite set is given independently of natural numbers:

166: 4234:

Call for an end to a discussion that seems to have reached a conclusion.

1543: 1508: 1482: 1141: 446: 4169:

says. It's the pieces that are isomorphic, not the balls themselves. --

1136: 4696: 4633: 4616: 4602: 4583: 4561: 4543: 4521: 4505: 4469: 4405: 4384: 4269: 4243: 4226: 4196: 4178: 4158: 4120: 4101: 3961: 3931: 3918: 3883: 3840: 3807: 3739: 3692: 3686: 3672: 3650: 3564: 3535: 3520: 3508: 3486: 3463: 3445: 3418: 3380: 3370: 3347: 3333: 3311: 3293: 3241: 3212: 3190: 3169: 3147: 3125: 3106: 3093: 3044: 3029: 3007: 2971: 2957: 2928: 2899: 2878: 2851: 2832: 2811: 2769: 2735: 2717: 2683: 2660: 2642: 2627: 2612: 2597: 2572: 2557: 2533: 2514: 2495: 2481: 2458: 2444: 2421: 2404: 2386: 2371: 2354: 2314: 2297: 2273: 2253: 2205: 2190: 2171: 2156: 2090: 2039: 2002: 1988: 1902: 1874: 1851: 1836: 1813: 1643: 1629: 1609: 1555: 1524: 1503: 1475: 1454: 1366: 1343: 1325: 1311: 1252: 1236: 1213: 1176: 1122: 1107: 1078: 1061: 1045: 1029: 1014: 994: 979: 964: 947: 928: 909: 893: 866: 850: 827: 806: 787: 770: 758: 743: 727: 700: 675: 659: 643: 628: 613: 598: 583: 553: 537: 521: 501: 472: 458: 431: 4504:

I see two obvious corrections, either all three concepts should share

4420:

Shouldn't this article spend more time on uses of the natural numbers?

3812:

By "ordered, countably-infinite sequence" I understand you to mean an

3691:"Natural numbers form a single concept" - citation needed. Certainly 3384: 2125:

As far as your "justifications" for "dropping zero", they sound like

1243:

A.Fraenkel 1968 (1953) Abstract set theory, North-Holland Amsterdam.

360: 1668:. International Series in Pure and Applied Mathematics (Third ed.). 1201:

Theorem: If a set A is finite, any set equipollent to A is finite.

4142: 4138: 3054: 3718:"it is much clearer to have a single article" - here we get into 4146: 3426:

What we have here is two essentially identical concepts with an

1515:. It is linked to in the article, and used at least four times. 2584:

I don't think we can simply say "0 is sometimes included". Per

2030:

In my experience, most books begin the natural numbers with 1.

1140:

An editor has identified a potential problem with the redirect

4260:. I guess a week is fine if no other supporters come forward. 2053:

to start them from 1, since nobody (but mathematicians) would

176: 168: 15: 4622:

too few for you to get the point, you are welcome to canvass

1357:

for explaining the relationship between the two approaches.

1150:

Knowledge:Redirects for discussion/Log/2022 July 28#N (math)

544:

set theory by constructing terms called finite ordinals. »?

4107:

relatively tangential to defining the natural numbers. Per

3677:

I agree. Chatul has made a strong case for not splitting.

3633:

one ball is isomorphic to two balls, are you going to make

2722:

Mathnerd, you seem to have the idea that this is an actual

2074:

is the one that contains the zeroth power of the variable).

1947:

Talk:Natural_number/Archive_3#Quotes_from_pre-algebra_books

2651:

should go into that level of detail, but not the lead. --

1922:

Talk:Natural_number/Archive_2#About_zero_as_natural_number

1480:

No, it is not. In fact, the article doesn't have the term

4642:

or that splitting the explanation across two pages would

4307: 3638: 3634: 2982: 2520:

even if he wasn't, if we use these terms, it sounds like

1800:

Note that cardinals and ordinals are discussed later, in

2760:

imply that such an opinion does not belong to the lead.

2319:

The issue is not stating the definitions. That is basic

1952:

Talk:Natural_number/Archive_3#Problems_with_this_article

1592:

That said, would you accept a footnote in the lead that

445:

Also, your formulation is wrong: "means formally" is an

4347: 4289: 4284: 3573:

You say they are "the same", but then they are written

2700:(emphasis mine): "NPOV policy means Knowledge editors 4039: 4000: 3853: 3780: 3753: 3607: 3579: 3479:

Mathnerd314159, you can't be serious with this one? –

2224: 1427: 1405: 1383: 3423:

Knowledge naming is not a strict popularity contest.

359:, a collaborative effort to improve the coverage of 273:, a collaborative effort to improve the coverage of 4375:for the reasons already expressed at-length above. 4084: 4008: 3862: 3786: 3762: 3621: 3593: 2232: 1433: 1413: 1391: 841:and the logical technicalities that they involve. 393:This article has not yet received a rating on the 1399:to be cardinals and ordinals only makes sense if 2288:older. But I guess I don't really know that. -- 1957:Talk:Natural_number/Archive_4#Where_do_we_start? 1916:Talk:Natural_number/Archive_1#Lack_of_references 33:for general discussion of the article's subject. 4731:Knowledge level-3 vital articles in Mathematics 4476:Article seems to confuse two different concepts 4290:the proposer of and only advocate for the split 3641:because "there is no mathematical difference"? 2544:does not make it explicit. Although checking a 2049:(that is, for counting and enumerating), it is 1942:Talk:Natural_number/Archive_2#Modern_Convention 2327:weight to one definition vs. another, using a 4085:{\displaystyle 0=\emptyset ,S(x)=x\cup \{x\}} 8: 4079: 4073: 2821:https://www.jstor.org/stable/pdf/3109881.pdf 1824:an ordinal is a property of natural numbers) 4257: 1789:, and numbers used for ordering are called 1781:in the first sentence the second sentence, 1225:Set-theoretic definition of natural numbers 970:What is the above comment responding to? -- 2910: 1967:Talk:Natural_number/Archive_4#Edit_request 324: 219: 4038: 4002: 4001: 3999: 3852: 3779: 3752: 3722:. It is certainly debatable but there is 3615: 3614: 3606: 3587: 3586: 3578: 3499:, which is established Knowledge policy. 3069:whereas in print the primary colours are 2702:ought to try to write sentences like this 2226: 2225: 2223: 1932:Talk:Natural_number/Archive_2#Conventions 1426: 1407: 1406: 1404: 1385: 1384: 1382: 4171:Shmuel (Seymour J.) Metz Username:Chatul 4094:Shmuel (Seymour J.) Metz Username:Chatul 3557:Shmuel (Seymour J.) Metz Username:Chatul 3140:Shmuel (Seymour J.) Metz Username:Chatul 2653:Shmuel (Seymour J.) Metz Username:Chatul 2397:Shmuel (Seymour J.) Metz Username:Chatul 2307:Shmuel (Seymour J.) Metz Username:Chatul 2246:Shmuel (Seymour J.) Metz Username:Chatul 1806:Shmuel (Seymour J.) Metz Username:Chatul 1636:Shmuel (Seymour J.) Metz Username:Chatul 1602:Shmuel (Seymour J.) Metz Username:Chatul 1496:Shmuel (Seymour J.) Metz Username:Chatul 1488:Shmuel (Seymour J.) Metz Username:Chatul 1462:This is already discussed at the end of 1447:Shmuel (Seymour J.) Metz Username:Chatul 1278:" is a pedantic way to say "the set has 818:(for exemple Fraenkel in Abstact Sets). 4721:Knowledge vital articles in Mathematics 4669: 1995:2A02:8070:6284:5100:F54B:87CF:FAC4:5CFB 1797:, is misleading and should be removed. 1656: 1353:. I have also added an introduction to 813:Existence of the set of natural numbers 326: 221: 180: 4351: 3035:black isn't strictly speaking a colour 3034: 1782: 1774: 877: 560: 4736:C-Class vital articles in Mathematics 4440:apart are considered equivalent; the 3944:here was never discussed, I checked. 1783:Numbers used for counting are called 706:I don't see why anyone would confuse 422:Intuitively, the natural Numbers … » 7: 4310:. Yellow to stay here, red to go to 3256:The following discussion is closed. 1666:Principles of Mathermatical Analysis 353:This article is within the scope of 267:This article is within the scope of 4756:Unknown-importance Numbers articles 3017: 1221:§ Constructions based on set theory 1219:This article has already a section 1094:, with the article "the" added and 210:It is of interest to the following 23:for discussing improvements to the 4046: 2178:Princeton Companion to Mathematics 1937:Talk:Natural_number/Archive_2#Zero 1911:Talk:Natural_number/Archive_1#Zero 1428: 1421:includes zero, since the null set 1316:I have added this to the article. 14: 4746:Top-priority mathematics articles 3622:{\displaystyle 1\in \mathbb {N} } 3594:{\displaystyle 0\in \mathbb {N} } 414:Formal and intuitive définitions. 287:Knowledge:WikiProject Mathematics 4716:Knowledge level-3 vital articles 4411:The discussion above is closed. 1377:Considering the natural numbers 1182:Constuctions based on set theory 1148:. This discussion will occur at 1135: 1085: 346: 328: 290:Template:WikiProject Mathematics 254: 244: 223: 190: 181: 150: 45:Click here to start a new topic. 3629:, clearly different. Under the 2045:If natural numbers are used as 307:This article has been rated as 4726:C-Class level-3 vital articles 4061: 4055: 1600:have more general meanings? -- 1494:) 14:33, 21 March 2023 (UTC)-- 1367:11:45, 21 September 2022 (UTC) 1344:07:46, 21 September 2022 (UTC) 1326:15:29, 20 September 2022 (UTC) 1312:15:17, 20 September 2022 (UTC) 1253:11:04, 20 September 2022 (UTC) 1237:17:13, 19 September 2022 (UTC) 1214:16:36, 19 September 2022 (UTC) 1123:07:31, 13 September 2022 (UTC) 1108:13:33, 12 September 2022 (UTC) 1079:11:40, 12 September 2022 (UTC) 980:21:08, 24 September 2021 (UTC) 965:08:51, 24 September 2021 (UTC) 948:18:04, 23 September 2021 (UTC) 929:17:55, 23 September 2021 (UTC) 910:17:27, 23 September 2021 (UTC) 894:16:39, 23 September 2021 (UTC) 867:15:30, 23 September 2021 (UTC) 851:13:46, 23 September 2021 (UTC) 828:13:23, 23 September 2021 (UTC) 807:11:53, 25 September 2021 (UTC) 788:10:58, 25 September 2021 (UTC) 771:20:56, 24 September 2021 (UTC) 759:17:24, 24 September 2021 (UTC) 744:16:39, 24 September 2021 (UTC) 728:19:58, 20 September 2021 (UTC) 701:06:10, 20 September 2021 (UTC) 676:20:20, 19 September 2021 (UTC) 660:15:51, 19 September 2021 (UTC) 644:08:52, 14 September 2021 (UTC) 629:13:39, 11 September 2021 (UTC) 614:11:39, 11 September 2021 (UTC) 599:11:23, 11 September 2021 (UTC) 584:09:53, 11 September 2021 (UTC) 554:09:24, 11 September 2021 (UTC) 538:11:50, 10 September 2021 (UTC) 522:11:00, 10 September 2021 (UTC) 502:10:43, 10 September 2021 (UTC) 473:11:39, 11 September 2021 (UTC) 1: 4494:Knowledge is not a dictionary 4352:it has not even been 24 hours 3900:here to build an encyclopedia 2579:Matrix (mathematics)#Notation 2459:19:02, 21 February 2024 (UTC) 2445:18:52, 21 February 2024 (UTC) 2405:18:18, 21 February 2024 (UTC) 2387:16:38, 21 February 2024 (UTC) 2372:11:07, 21 February 2024 (UTC) 2355:20:44, 20 February 2024 (UTC) 2315:14:36, 20 February 2024 (UTC) 2298:17:43, 20 February 2024 (UTC) 2274:11:36, 20 February 2024 (UTC) 2254:11:08, 20 February 2024 (UTC) 2206:20:47, 19 February 2024 (UTC) 2191:19:43, 19 February 2024 (UTC) 2172:17:43, 19 February 2024 (UTC) 2157:17:28, 19 February 2024 (UTC) 2091:15:40, 19 February 2024 (UTC) 2040:11:35, 19 February 2024 (UTC) 1989:01:01, 19 February 2024 (UTC) 1903:00:24, 19 February 2024 (UTC) 1875:20:38, 18 February 2024 (UTC) 1773:reverted it with the comment 1062:08:18, 9 September 2022 (UTC) 1046:11:25, 8 September 2022 (UTC) 1030:10:54, 8 September 2022 (UTC) 1015:10:30, 7 September 2022 (UTC) 995:08:22, 7 September 2022 (UTC) 459:11:54, 9 September 2021 (UTC) 432:11:18, 9 September 2021 (UTC) 373:Knowledge:WikiProject Numbers 367:and see a list of open tasks. 281:and see a list of open tasks. 42:Put new text under old text. 4761:WikiProject Numbers articles 4741:C-Class mathematics articles 4306:Here is the proposed split: 4009:{\displaystyle \mathbb {N} } 2233:{\displaystyle \mathbb {N} } 2218:There are two approaches to 1414:{\displaystyle \mathbb {N} } 1392:{\displaystyle \mathbb {N} } 1373:Problem with starting from 1 376:Template:WikiProject Numbers 4697:20:17, 12 August 2024 (UTC) 3276:As justification, there is 1693:"Formal Logic - Set Theory" 50:New to Knowledge? Welcome! 4777: 4484:Further, this article and 2770:09:35, 20 March 2024 (UTC) 2736:23:21, 19 March 2024 (UTC) 2718:23:13, 19 March 2024 (UTC) 2684:17:57, 19 March 2024 (UTC) 2661:12:38, 19 March 2024 (UTC) 2643:00:52, 19 March 2024 (UTC) 2628:23:59, 18 March 2024 (UTC) 2613:22:50, 18 March 2024 (UTC) 2598:21:08, 18 March 2024 (UTC) 2573:20:42, 18 March 2024 (UTC) 2558:20:04, 18 March 2024 (UTC) 2534:18:20, 18 March 2024 (UTC) 2515:17:48, 18 March 2024 (UTC) 2496:17:29, 18 March 2024 (UTC) 2482:01:20, 17 March 2024 (UTC) 1852:17:58, 21 March 2023 (UTC) 1837:16:04, 21 March 2023 (UTC) 1814:14:44, 21 March 2023 (UTC) 1644:13:41, 18 March 2024 (UTC) 1630:07:11, 17 March 2024 (UTC) 1610:20:05, 22 March 2023 (UTC) 1556:16:58, 21 March 2023 (UTC) 1525:15:39, 21 March 2023 (UTC) 1504:14:33, 21 March 2023 (UTC) 1476:12:29, 21 March 2023 (UTC) 1455:12:21, 21 March 2023 (UTC) 1434:{\displaystyle \emptyset } 1351:§ Set-theoretic definition 839:foundations of mathematics 395:project's importance scale 4655:16:56, 18 June 2024 (UTC) 4634:07:06, 18 June 2024 (UTC) 4617:05:18, 18 June 2024 (UTC) 4603:04:40, 18 June 2024 (UTC) 4584:01:33, 18 June 2024 (UTC) 4562:06:30, 17 June 2024 (UTC) 4544:05:57, 17 June 2024 (UTC) 4522:05:37, 17 June 2024 (UTC) 4165:No, that is not what the 3300:Strongest possible oppose 2900:17:34, 7 April 2024 (UTC) 2879:04:20, 7 April 2024 (UTC) 2706:Van der Waerden's theorem 2009:0 is not a natural number 1887:0 is not a natural number 1777:. I believe that without 1177:03:57, 28 July 2022 (UTC) 392: 341: 306: 239: 218: 80:Be welcoming to newcomers 4751:C-Class Numbers articles 4470:03:42, 6 June 2024 (UTC) 4455:02:40, 6 June 2024 (UTC) 4413:Please do not modify it. 4406:03:36, 6 June 2024 (UTC) 4385:00:30, 6 June 2024 (UTC) 4364:00:24, 6 June 2024 (UTC) 4343:23:56, 5 June 2024 (UTC) 4328:23:38, 5 June 2024 (UTC) 4302:23:18, 5 June 2024 (UTC) 4270:18:05, 5 June 2024 (UTC) 4244:10:07, 5 June 2024 (UTC) 4227:02:48, 5 June 2024 (UTC) 4197:18:00, 5 June 2024 (UTC) 4179:16:32, 5 June 2024 (UTC) 4159:17:46, 5 June 2024 (UTC) 4121:18:28, 5 June 2024 (UTC) 4102:16:32, 5 June 2024 (UTC) 3976:00:09, 6 June 2024 (UTC) 3962:23:03, 5 June 2024 (UTC) 3932:19:50, 5 June 2024 (UTC) 3919:18:19, 5 June 2024 (UTC) 3884:07:14, 5 June 2024 (UTC) 3863:{\displaystyle -4{,}321} 3841:06:38, 5 June 2024 (UTC) 3808:05:56, 5 June 2024 (UTC) 3740:03:22, 5 June 2024 (UTC) 3687:01:03, 5 June 2024 (UTC) 3673:21:22, 4 June 2024 (UTC) 3651:20:32, 4 June 2024 (UTC) 3565:17:32, 4 June 2024 (UTC) 3536:06:57, 5 June 2024 (UTC) 3521:22:22, 4 June 2024 (UTC) 3509:20:32, 4 June 2024 (UTC) 3487:14:49, 4 June 2024 (UTC) 3464:06:48, 5 June 2024 (UTC) 3446:02:36, 5 June 2024 (UTC) 3419:20:32, 4 June 2024 (UTC) 3371:09:01, 4 June 2024 (UTC) 3348:22:14, 4 June 2024 (UTC) 3334:20:32, 4 June 2024 (UTC) 3312:07:33, 4 June 2024 (UTC) 3294:05:19, 4 June 2024 (UTC) 3258:Please do not modify it. 3242:17:35, 6 June 2024 (UTC) 3213:01:44, 5 June 2024 (UTC) 3191:00:59, 5 June 2024 (UTC) 3170:13:06, 7 June 2024 (UTC) 3148:12:36, 7 June 2024 (UTC) 3126:21:06, 6 June 2024 (UTC) 3107:20:30, 6 June 2024 (UTC) 3094:20:15, 6 June 2024 (UTC) 3045:07:20, 5 June 2024 (UTC) 3030:02:45, 5 June 2024 (UTC) 3018:the split proposal below 3008:23:20, 4 June 2024 (UTC) 2972:05:19, 4 June 2024 (UTC) 2958:05:04, 4 June 2024 (UTC) 2929:00:51, 4 June 2024 (UTC) 2852:01:12, 5 June 2024 (UTC) 2833:00:37, 5 June 2024 (UTC) 2812:23:39, 4 June 2024 (UTC) 2692:policy, "This policy is 2422:01:18, 4 June 2024 (UTC) 2176:To quote page 17 of the 2145:recently-published books 2003:13:08, 24 May 2024 (UTC) 1260:§ von Neumann definition 1130:Redirects for discussion 313:project's priority scale 4258:#Is 0 a natural number? 2688:There is a line in the 1290:elements if there is a 1266:is defined as a set of 1144:and has thus listed it 270:WikiProject Mathematics 4711:C-Class vital articles 4308:https://ibb.co/3hgLRTS 4086: 4010: 3896:not writing a textbook 3864: 3818:arithmetic progression 3788: 3764: 3623: 3595: 3061:(light) as opposed to 2391:What's in Book VII of 2234: 1907:Previous discussions: 1883:a matter of definition 1860:Is 0 a natural number? 1664:Rudin, Walter (1976). 1435: 1415: 1393: 835:set of natural numbers 75:avoid personal attacks 4640:two separate concepts 4167:Banach–Tarski theorem 4087: 4011: 3865: 3789: 3765: 3631:Banach–Tarski paradox 3624: 3596: 2502:WP:TECHNICAL#Audience 2235: 2007:I would not consider 1436: 1416: 1394: 1128:"N (math)" listed at 197:level-3 vital article 144:Auto-archiving period 100:Neutral point of view 4312:non-negative integer 4037: 3998: 3942:Non-negative integer 3851: 3822:consecutive integers 3778: 3751: 3605: 3577: 3271:Non-negative integer 2541:universally accepted 2331:tone and avoiding a 2222: 2136:matter of definition 1464:§ Modern definitions 1425: 1403: 1381: 1355:§ Formal definitions 795:§ Modern definitions 716:intuitionistic logic 689:Intuitionistic logic 572:§ Formal definitions 293:mathematics articles 105:No original research 4527:<tired sigh: --> 4442:Eisenstein integers 4028:Raphael M. Robinson 3696:deliberately vague. 3431:here in my opinion. 2647:I believe that the 2118:As for the symbol, 1571:as synonymous with 356:WikiProject Numbers 4644:help anyone learn. 4550:Usina do Gasômetro 4082: 4006: 3988:No, there are not 3907:countably infinite 3860: 3816:, specifically an 3784: 3763:{\displaystyle -1} 3760: 3619: 3591: 3282:different articles 3259: 2962:edit: new section 2409:The definition in 2230: 1431: 1411: 1389: 262:Mathematics portal 206:content assessment 86:dispute resolution 47: 4492:Given this fact, 3820:or a sequence of 3787:{\displaystyle 3} 3322:Ariane flight V88 3257: 3250: 3247:non-admin closure 3134:Actually, there ' 2931: 2915:comment added by 2393:Euclid's Elements 2362:Giuseppe Peano;" 1765:I added the word 1558: 1539: 1536: 512:your suggestion. 409: 408: 405: 404: 401: 400: 323: 322: 319: 318: 175: 174: 66:Assume good faith 43: 4768: 4681: 4674: 4316:positive integer 4314:, blue to go to 4091: 4089: 4088: 4083: 4024:John von Neumann 4020:Peano postulates 4017: 4015: 4013: 4012: 4007: 4005: 3938:Positive integer 3871: 3869: 3867: 3866: 3861: 3814:integer sequence 3795: 3793: 3791: 3790: 3785: 3771: 3769: 3767: 3766: 3761: 3628: 3626: 3625: 3620: 3618: 3600: 3598: 3597: 3592: 3590: 3553:Peano postulates 3452:integer sequence 3428:off-by-one error 3267:Positive integer 3244: 2987:WP:NOTDICTIONARY 2945:WP:NOTDICTIONARY 2239: 2237: 2236: 2231: 2229: 2059:cardinal numbers 1891:WP:Broad concept 1802:#Generalizations 1786:cardinal numbers 1753: 1752: 1749:Wolfram Research 1735: 1729: 1728: 1725:Wolfram Research 1711: 1705: 1704: 1689: 1683: 1682: 1661: 1541: 1538: 1529: 1440: 1438: 1437: 1432: 1420: 1418: 1417: 1412: 1410: 1398: 1396: 1395: 1390: 1388: 1301: 1297: 1289: 1285: 1281: 1277: 1269: 1265: 1173: 1168: 1139: 1093: 1089: 1088: 763:And me as well. 381: 380: 379:Numbers articles 377: 374: 371: 350: 343: 342: 332: 325: 295: 294: 291: 288: 285: 264: 259: 258: 248: 241: 240: 235: 227: 220: 203: 194: 193: 186: 185: 177: 169: 155: 154: 145: 95:Article policies 16: 4776: 4775: 4771: 4770: 4769: 4767: 4766: 4765: 4701: 4700: 4685: 4684: 4675: 4671: 4478: 4422: 4417: 4416: 4279:with a side of 4232:Strongly Oppose 4035: 4034: 3996: 3995: 3993: 3849: 3848: 3846: 3776: 3775: 3773: 3749: 3748: 3746: 3679:Montgomery Link 3603: 3602: 3575: 3574: 3262: 3253: 3252: 3251: 3227: 3197:Montgomery Link 3183:Montgomery Link 2917:183.182.115.217 2839:Montgomery Link 2825:Montgomery Link 2796:Montgomery Link 2474:Montgomery Link 2337:Montgomery Link 2333:WP:FALSEBALANCE 2220: 2219: 2131:no common sense 2047:ordinal numbers 1862: 1792:ordinal numbers 1763: 1758: 1757: 1756: 1737: 1736: 1732: 1713: 1712: 1708: 1691: 1690: 1686: 1679: 1663: 1662: 1658: 1423: 1422: 1401: 1400: 1379: 1378: 1375: 1299: 1295: 1287: 1283: 1279: 1275: 1267: 1263: 1223:with a link to 1184: 1171: 1166: 1133: 1086: 1084: 815: 510:strongly oppose 492:The set of … » 416: 378: 375: 372: 369: 368: 292: 289: 286: 283: 282: 260: 253: 233: 204:on Knowledge's 201: 191: 171: 170: 165: 142: 116: 115: 114: 91: 61: 12: 11: 5: 4774: 4772: 4764: 4763: 4758: 4753: 4748: 4743: 4738: 4733: 4728: 4723: 4718: 4713: 4703: 4702: 4683: 4682: 4668: 4667: 4666: 4665: 4664: 4663: 4662: 4661: 4660: 4659: 4658: 4657: 4636: 4619: 4595:Mathnerd314159 4590: 4567: 4554:Mathnerd314159 4532: 4528: 4477: 4474: 4473: 4472: 4421: 4418: 4410: 4409: 4408: 4387: 4370: 4369: 4368: 4367: 4366: 4345: 4320:Mathnerd314159 4274: 4273: 4272: 4262:Mathnerd314159 4254: 4229: 4206: 4205: 4204: 4203: 4202: 4201: 4200: 4199: 4189:Mathnerd314159 4185: 4163: 4162: 4161: 4151:Mathnerd314159 4131: 4130: 4129: 4128: 4127: 4126: 4125: 4124: 4123: 4113:Mathnerd314159 4081: 4078: 4075: 4072: 4069: 4066: 4063: 4060: 4057: 4054: 4051: 4048: 4045: 4042: 4031: 4004: 3986: 3985: 3984: 3983: 3982: 3981: 3980: 3979: 3978: 3954:Mathnerd314159 3949: 3911:Mathnerd314159 3903: 3892: 3889: 3873: 3859: 3856: 3833:Mathnerd314159 3828: 3825: 3797: 3783: 3759: 3756: 3732:Mathnerd314159 3716: 3705: 3697: 3689: 3643:Mathnerd314159 3617: 3613: 3610: 3589: 3585: 3582: 3568: 3567: 3545: 3544: 3543: 3542: 3541: 3540: 3539: 3538: 3528:Mathnerd314159 3501:Mathnerd314159 3490: 3489: 3473: 3472: 3471: 3470: 3469: 3468: 3467: 3466: 3456:Mathnerd314159 3435: 3432: 3424: 3411:Mathnerd314159 3393:complex number 3374: 3373: 3355: 3354: 3353: 3352: 3351: 3350: 3326:Mathnerd314159 3315: 3314: 3286:Mathnerd314159 3263: 3254: 3231: 3230: 3229: 3228: 3226: 3225:Split proposal 3223: 3222: 3221: 3220: 3219: 3218: 3217: 3216: 3215: 3201: 3178: 3177: 3176: 3175: 3174: 3173: 3172: 3158: 3132: 3131: 3130: 3129: 3128: 3082: 3078: 3074: 3032: 3022:Mathnerd314159 2995: 2991: 2978:Mathnerd314159 2974: 2964:Mathnerd314159 2950:Mathnerd314159 2940: 2936: 2905: 2904: 2903: 2902: 2888: 2871:Mathnerd314159 2863: 2862: 2861: 2860: 2859: 2858: 2857: 2856: 2855: 2854: 2817: 2801: 2800: 2799: 2792: 2791: 2790: 2789: 2788: 2787: 2786: 2785: 2784: 2783: 2782: 2781: 2780: 2779: 2778: 2777: 2776: 2775: 2774: 2773: 2772: 2738: 2710:Mathnerd314159 2694:non-negotiable 2671: 2620:Mathnerd314159 2590:Mathnerd314159 2582: 2550:Mathnerd314159 2507:Mathnerd314159 2498: 2488:Mathnerd314159 2469: 2468: 2467: 2466: 2465: 2464: 2463: 2462: 2461: 2437:Mathnerd314159 2426: 2425: 2424: 2347:Mathnerd314159 2317: 2302: 2301: 2300: 2257: 2256: 2228: 2215: 2214: 2213: 2212: 2211: 2210: 2209: 2208: 2198:Mathnerd314159 2164:Mathnerd314159 2149:Mathnerd314159 2140: 2123: 2116: 2093: 2075: 2020: 2019: 2018: 2017: 1991: 1981:Mathnerd314159 1976: 1975: 1974: 1969: 1964: 1959: 1954: 1949: 1944: 1939: 1934: 1929: 1924: 1919: 1913: 1905: 1895:Mathnerd314159 1861: 1858: 1857: 1856: 1855: 1854: 1844:Mathnerd314159 1825: 1762: 1759: 1755: 1754: 1730: 1706: 1684: 1677: 1655: 1654: 1650: 1649: 1648: 1647: 1646: 1618: 1617: 1616: 1615: 1614: 1613: 1612: 1583: 1582: 1581: 1580: 1579: 1578: 1577: 1576: 1561: 1559: 1537: 1460: 1430: 1409: 1387: 1374: 1371: 1370: 1369: 1331: 1330: 1329: 1328: 1272: 1240: 1239: 1183: 1180: 1146:for discussion 1132: 1126: 1111: 1110: 1049: 1048: 1018: 1017: 1002: 983: 982: 953: 952: 951: 950: 897: 896: 881: 874: 854: 853: 814: 811: 810: 809: 776: 775: 774: 773: 731: 730: 681: 680: 679: 678: 617: 616: 587: 586: 563:Peano's axioms 541: 540: 524: 476: 475: 461: 443: 415: 412: 407: 406: 403: 402: 399: 398: 391: 385: 384: 382: 365:the discussion 351: 339: 338: 333: 321: 320: 317: 316: 305: 299: 298: 296: 279:the discussion 266: 265: 249: 237: 236: 228: 216: 215: 209: 187: 173: 172: 163: 161: 160: 157: 156: 118: 117: 113: 112: 107: 102: 93: 92: 90: 89: 82: 77: 68: 62: 60: 59: 48: 39: 38: 35: 34: 28: 25:Natural number 13: 10: 9: 6: 4: 3: 2: 4773: 4762: 4759: 4757: 4754: 4752: 4749: 4747: 4744: 4742: 4739: 4737: 4734: 4732: 4729: 4727: 4724: 4722: 4719: 4717: 4714: 4712: 4709: 4708: 4706: 4699: 4698: 4694: 4690: 4689:175.100.6.159 4679: 4673: 4670: 4656: 4652: 4648: 4645: 4641: 4637: 4635: 4632: 4629: 4625: 4620: 4618: 4614: 4610: 4606: 4605: 4604: 4600: 4596: 4591: 4587: 4586: 4585: 4581: 4577: 4573: 4568: 4565: 4564: 4563: 4559: 4555: 4551: 4547: 4546: 4545: 4541: 4537: 4533: 4529: 4526: 4525: 4524: 4523: 4519: 4515: 4511: 4507: 4502: 4498: 4495: 4490: 4487: 4482: 4475: 4471: 4467: 4463: 4459: 4458: 4457: 4456: 4453: 4450: 4445: 4443: 4439: 4435: 4429: 4425: 4414: 4407: 4403: 4399: 4395: 4391: 4388: 4386: 4382: 4378: 4374: 4371: 4365: 4361: 4357: 4353: 4349: 4346: 4344: 4340: 4336: 4331: 4330: 4329: 4325: 4321: 4317: 4313: 4309: 4305: 4304: 4303: 4299: 4295: 4291: 4286: 4282: 4278: 4277:Strong Oppose 4275: 4271: 4267: 4263: 4259: 4255: 4251: 4247: 4246: 4245: 4241: 4237: 4233: 4230: 4228: 4224: 4220: 4216: 4211: 4208: 4207: 4198: 4194: 4190: 4186: 4182: 4181: 4180: 4176: 4172: 4168: 4164: 4160: 4156: 4152: 4148: 4144: 4140: 4136: 4132: 4122: 4118: 4114: 4110: 4105: 4104: 4103: 4099: 4095: 4076: 4070: 4067: 4064: 4058: 4052: 4049: 4043: 4040: 4032: 4029: 4025: 4021: 3991: 3990:precisely two 3987: 3977: 3973: 3969: 3965: 3964: 3963: 3959: 3955: 3950: 3947: 3943: 3939: 3935: 3934: 3933: 3930: 3927: 3922: 3921: 3920: 3916: 3912: 3908: 3904: 3901: 3897: 3893: 3890: 3887: 3886: 3885: 3882: 3879: 3874: 3857: 3854: 3844: 3843: 3842: 3838: 3834: 3829: 3826: 3823: 3819: 3815: 3811: 3810: 3809: 3806: 3803: 3798: 3781: 3757: 3754: 3743: 3742: 3741: 3737: 3733: 3729: 3728:even if short 3725: 3721: 3717: 3714: 3710: 3706: 3703: 3698: 3694: 3690: 3688: 3684: 3680: 3676: 3675: 3674: 3670: 3666: 3662: 3658: 3654: 3653: 3652: 3648: 3644: 3640: 3636: 3632: 3611: 3608: 3583: 3580: 3572: 3571: 3570: 3569: 3566: 3562: 3558: 3554: 3550: 3547: 3546: 3537: 3533: 3529: 3524: 3523: 3522: 3519: 3516: 3512: 3511: 3510: 3506: 3502: 3498: 3494: 3493: 3492: 3491: 3488: 3485: 3482: 3478: 3475: 3474: 3465: 3461: 3457: 3453: 3449: 3448: 3447: 3444: 3441: 3436: 3433: 3429: 3425: 3422: 3421: 3420: 3416: 3412: 3408: 3404: 3401: 3397: 3394: 3390: 3386: 3382: 3378: 3377: 3376: 3375: 3372: 3368: 3364: 3360: 3357: 3356: 3349: 3345: 3341: 3337: 3336: 3335: 3331: 3327: 3323: 3319: 3318: 3317: 3316: 3313: 3309: 3305: 3301: 3298: 3297: 3296: 3295: 3291: 3287: 3283: 3279: 3274: 3272: 3268: 3261: 3248: 3243: 3239: 3235: 3224: 3214: 3210: 3206: 3202: 3198: 3194: 3193: 3192: 3188: 3184: 3179: 3171: 3167: 3163: 3159: 3155: 3151: 3150: 3149: 3145: 3141: 3137: 3133: 3127: 3123: 3119: 3114: 3110: 3109: 3108: 3105: 3102: 3097: 3096: 3095: 3091: 3087: 3083: 3079: 3075: 3072: 3068: 3064: 3060: 3056: 3052: 3048: 3047: 3046: 3043: 3040: 3036: 3033: 3031: 3027: 3023: 3019: 3015: 3011: 3010: 3009: 3005: 3001: 2996: 2992: 2988: 2984: 2979: 2975: 2973: 2969: 2965: 2961: 2960: 2959: 2955: 2951: 2946: 2941: 2937: 2933: 2932: 2930: 2926: 2922: 2918: 2914: 2907: 2906: 2901: 2897: 2893: 2889: 2885: 2884: 2883: 2882: 2881: 2880: 2876: 2872: 2867: 2853: 2849: 2845: 2840: 2836: 2835: 2834: 2830: 2826: 2822: 2818: 2815: 2814: 2813: 2809: 2805: 2802: 2797: 2793: 2771: 2767: 2763: 2759: 2755: 2751: 2747: 2743: 2739: 2737: 2733: 2729: 2725: 2721: 2720: 2719: 2715: 2711: 2707: 2703: 2699: 2695: 2691: 2687: 2686: 2685: 2681: 2677: 2672: 2668: 2664: 2663: 2662: 2658: 2654: 2650: 2646: 2645: 2644: 2640: 2636: 2631: 2630: 2629: 2625: 2621: 2616: 2615: 2614: 2610: 2606: 2601: 2600: 2599: 2595: 2591: 2587: 2583: 2580: 2576: 2575: 2574: 2570: 2566: 2561: 2560: 2559: 2555: 2551: 2547: 2542: 2537: 2536: 2535: 2531: 2527: 2523: 2518: 2517: 2516: 2512: 2508: 2503: 2499: 2497: 2493: 2489: 2485: 2484: 2483: 2479: 2475: 2470: 2460: 2456: 2452: 2448: 2447: 2446: 2442: 2438: 2434: 2431: 2427: 2423: 2420: 2417: 2412: 2408: 2407: 2406: 2402: 2398: 2394: 2390: 2389: 2388: 2384: 2380: 2375: 2374: 2373: 2369: 2365: 2360: 2359: 2358: 2357: 2356: 2352: 2348: 2343: 2338: 2334: 2330: 2326: 2322: 2318: 2316: 2312: 2308: 2303: 2299: 2295: 2291: 2287: 2286: 2280: 2279: 2278: 2277: 2276: 2275: 2271: 2267: 2261: 2255: 2251: 2247: 2243: 2217: 2216: 2207: 2203: 2199: 2194: 2193: 2192: 2188: 2184: 2179: 2175: 2174: 2173: 2169: 2165: 2160: 2159: 2158: 2154: 2150: 2146: 2141: 2137: 2134:for 1 in the 2132: 2128: 2124: 2121: 2117: 2114: 2112: 2109: 2107: 2105: 2103: 2101: 2098: 2094: 2092: 2088: 2084: 2080: 2076: 2073: 2069: 2064: 2060: 2056: 2052: 2048: 2044: 2043: 2042: 2041: 2037: 2033: 2028: 2024: 2014: 2010: 2006: 2005: 2004: 2000: 1996: 1992: 1990: 1986: 1982: 1977: 1973: 1970: 1968: 1965: 1963: 1960: 1958: 1955: 1953: 1950: 1948: 1945: 1943: 1940: 1938: 1935: 1933: 1930: 1928: 1925: 1923: 1920: 1917: 1914: 1912: 1909: 1908: 1906: 1904: 1900: 1896: 1892: 1888: 1884: 1879: 1878: 1877: 1876: 1872: 1868: 1859: 1853: 1849: 1845: 1840: 1839: 1838: 1834: 1830: 1826: 1823: 1818: 1817: 1816: 1815: 1811: 1807: 1803: 1798: 1796: 1794: 1793: 1788: 1787: 1780: 1776: 1772: 1768: 1760: 1750: 1746: 1745: 1740: 1734: 1731: 1726: 1722: 1721: 1716: 1710: 1707: 1703: 1698: 1694: 1688: 1685: 1680: 1678:0-07-054235-X 1675: 1671: 1667: 1660: 1657: 1653: 1645: 1641: 1637: 1633: 1632: 1631: 1627: 1623: 1619: 1611: 1607: 1603: 1599: 1595: 1591: 1590: 1589: 1588: 1587: 1586: 1585: 1584: 1574: 1570: 1566: 1562: 1560: 1557: 1553: 1549: 1545: 1540: 1534: 1528: 1527: 1526: 1522: 1518: 1514: 1510: 1507: 1506: 1505: 1501: 1497: 1493: 1489: 1485: 1484: 1479: 1478: 1477: 1473: 1469: 1465: 1461: 1459: 1458: 1457: 1456: 1452: 1448: 1444: 1372: 1368: 1364: 1360: 1356: 1352: 1348: 1347: 1346: 1345: 1341: 1337: 1327: 1323: 1319: 1315: 1314: 1313: 1309: 1305: 1293: 1273: 1261: 1257: 1256: 1255: 1254: 1250: 1246: 1238: 1234: 1230: 1226: 1222: 1218: 1217: 1216: 1215: 1211: 1207: 1202: 1199: 1196: 1193: 1190: 1187: 1181: 1179: 1178: 1174: 1169: 1163: 1162: 1159: 1156: 1151: 1147: 1143: 1138: 1131: 1127: 1125: 1124: 1120: 1116: 1109: 1105: 1101: 1097: 1092: 1083: 1082: 1081: 1080: 1076: 1072: 1067: 1064: 1063: 1059: 1055: 1047: 1043: 1039: 1034: 1033: 1032: 1031: 1027: 1023: 1016: 1012: 1008: 1003: 999: 998: 997: 996: 992: 988: 981: 977: 973: 969: 968: 967: 966: 962: 958: 949: 945: 941: 937: 932: 931: 930: 926: 922: 918: 914: 913: 912: 911: 907: 903: 895: 891: 887: 882: 879: 875: 871: 870: 869: 868: 864: 860: 852: 848: 844: 840: 836: 832: 831: 830: 829: 825: 821: 812: 808: 804: 800: 796: 792: 791: 790: 789: 785: 781: 772: 769: 766: 762: 761: 760: 756: 752: 748: 747: 746: 745: 741: 737: 729: 725: 721: 717: 713: 709: 705: 704: 703: 702: 698: 694: 690: 686: 677: 674: 671: 667: 664:I agree with 663: 662: 661: 657: 653: 648: 647: 646: 645: 641: 637: 631: 630: 626: 622: 615: 611: 607: 603: 602: 601: 600: 596: 592: 585: 581: 577: 573: 569: 568: 564: 558: 557: 556: 555: 551: 547: 539: 536: 533: 529: 526:I agree with 525: 523: 519: 515: 511: 506: 505: 504: 503: 499: 495: 490: 487: 484: 480: 474: 470: 466: 462: 460: 456: 452: 448: 444: 440: 436: 435: 434: 433: 429: 425: 420: 413: 411: 396: 390: 387: 386: 383: 366: 362: 358: 357: 352: 349: 345: 344: 340: 337: 334: 331: 327: 314: 310: 304: 301: 300: 297: 280: 276: 272: 271: 263: 257: 252: 250: 247: 243: 242: 238: 232: 229: 226: 222: 217: 213: 207: 199: 198: 188: 184: 179: 178: 159: 158: 153: 149: 141: 137: 133: 129: 126: 124: 120: 119: 111: 110:Verifiability 108: 106: 103: 101: 98: 97: 96: 87: 83: 81: 78: 76: 72: 69: 67: 64: 63: 57: 53: 52:Learn to edit 49: 46: 41: 40: 37: 36: 32: 26: 22: 18: 17: 4686: 4677: 4672: 4643: 4639: 4571: 4510:Whole number 4503: 4499: 4491: 4486:Whole number 4483: 4479: 4446: 4437: 4433: 4430: 4426: 4423: 4412: 4389: 4372: 4285:tried a SNOW 4280: 4276: 4236:Rick Norwood 4231: 4215:Prime number 4209: 4109:WP:TECHNICAL 3989: 3727: 3709:whole number 3661:Peano axioms 3637:redirect to 3548: 3476: 3407:whole number 3358: 3299: 3281: 3275: 3264: 3255: 3135: 2911:— Preceding 2892:Rick Norwood 2868: 2864: 2723: 2701: 2693: 2666: 2648: 2521: 2410: 2364:Rick Norwood 2342:WP:CONSENSUS 2329:WP:IMPARTIAL 2284: 2283: 2266:Rick Norwood 2262: 2258: 2097:Rick Norwood 2078: 2072:power series 2067: 2062: 2054: 2050: 2032:Rick Norwood 2029: 2025: 2021: 1863: 1821: 1799: 1790: 1784: 1778: 1766: 1764: 1742: 1733: 1718: 1709: 1700: 1687: 1665: 1659: 1651: 1597: 1593: 1572: 1568: 1564: 1532: 1481: 1376: 1332: 1241: 1203: 1200: 1197: 1194: 1191: 1188: 1185: 1160: 1157: 1154: 1134: 1112: 1090: 1068: 1065: 1050: 1019: 984: 954: 935: 898: 855: 834: 816: 777: 732: 715: 712:intuitionism 711: 707: 685:Intuitionism 682: 632: 618: 588: 542: 509: 491: 488: 485: 481: 477: 439:WP:TECHNICAL 421: 417: 410: 354: 309:Top-priority 308: 268: 234:Top‑priority 212:WikiProjects 195: 147: 121: 94: 19:This is the 4350:, you said 4281:touch grass 3724:WP:NOTMERGE 3702:Chris Ofili 3389:real number 3063:subtractive 2577:Looking at 2129:. There is 2016:definition. 1715:"Empty Set" 1670:McGraw Hill 765:Paul August 708:intuitively 693:SilverMatsu 670:Paul August 532:Paul August 284:Mathematics 275:mathematics 231:Mathematics 31:not a forum 4705:Categories 4687:066239319 4647:XOR'easter 4377:Walsh90210 4356:XOR'easter 4335:XOR'easter 4294:XOR'easter 4184:structure. 4018:using the 3968:XOR'easter 3657:WP:NOTDICT 3497:WP:NOTDICT 3278:WP:NOTDICT 3014:WP:NOTDICT 2605:Popcountll 2546:comparison 1918:(tail end) 1739:"Null Set" 1652:References 1486:at all. -- 1096:number set 567:set theory 4628:jacobolus 4609:Trovatore 4576:Trovatore 4536:Trovatore 4531:included. 4449:jacobolus 4250:WP:STRONG 4248:There is 3926:jacobolus 3878:jacobolus 3802:jacobolus 3515:jacobolus 3481:jacobolus 3440:jacobolus 3340:Trovatore 3304:Trovatore 3113:Jacobolus 3101:jacobolus 3051:Jacobolus 3039:jacobolus 2728:Trovatore 2676:Trovatore 2635:Trovatore 2526:Trovatore 2451:Trovatore 2416:jacobolus 2379:Trovatore 2290:Trovatore 2242:empty set 2055:naturally 1885:and that 1744:MathWorld 1720:MathWorld 1697:Britanica 1548:Trovatore 1513:empty set 1292:bijection 1271:ordinals. 972:Trovatore 940:Trovatore 720:Anita5192 200:is rated 88:if needed 71:Be polite 21:talk page 4572:pick one 4514:Awwright 4394:WP:CFORK 4253:strong." 4135:D.Lazard 3720:WP:MERGE 3713:proposal 3665:D.Lazard 3363:D.Lazard 3059:additive 2994:heading? 2925:contribs 2913:unsigned 2762:D.Lazard 2698:WP:RNPOV 2565:D.Lazard 2430:MacTutor 2411:Elements 2325:WP:UNDUE 2139:sources. 2083:D.Lazard 2061:, it is 2011:to be a 1829:D.Lazard 1771:D.Lazard 1594:cardinal 1569:null set 1565:null set 1544:New Math 1517:D.Lazard 1509:Null set 1483:null set 1468:D.Lazard 1359:D.Lazard 1336:CBerlioz 1318:D.Lazard 1304:D.Lazard 1245:CBerlioz 1229:D.Lazard 1206:CBerlioz 1142:N (math) 1115:CBerlioz 1100:D.Lazard 1071:CBerlioz 1054:CBerlioz 1038:D.Lazard 1022:CBerlioz 1007:D.Lazard 987:CBerlioz 957:CBerlioz 921:D.Lazard 902:CBerlioz 886:D.Lazard 859:CBerlioz 843:D.Lazard 820:CBerlioz 799:D.Lazard 780:CBerlioz 751:D.Lazard 736:CBerlioz 666:D.Lazard 652:D.Lazard 636:CBerlioz 621:CBerlioz 606:D.Lazard 591:CBerlioz 576:D.Lazard 546:CBerlioz 528:D.Lazard 514:D.Lazard 494:CBerlioz 465:D.Lazard 451:D.Lazard 447:oxymoron 424:CBerlioz 148:365 days 123:Archives 56:get help 29:This is 27:article. 4678:someone 4506:Integer 3946:Silence 3693:concept 3400:WikiNav 3381:integer 3205:SaintIX 3162:SaintIX 3118:SaintIX 3086:SaintIX 3000:SaintIX 2844:SaintIX 2804:SaintIX 2758:WP:NPOV 2750:WP:NPOV 2742:WP:NPOV 2724:dispute 2690:WP:NPOV 2649:article 2586:WP:NPOV 2321:WP:NPOV 2079:support 2068:natural 2063:natural 2051:natural 1622:Ztimes3 1598:ordinal 1155:Laundry 1098:linked 873:theory. 370:Numbers 361:Numbers 336:Numbers 311:on the 202:C-class 4624:WT:WPM 4390:Oppose 4373:Oppose 4219:Meters 4210:Oppose 3898:. I'm 3549:OPPOSE 3477:Oppose 3385:number 3359:Oppose 3154:Chatul 3055:colour 2754:WP:NOR 2077:So, I 1867:Certes 1779:finite 1767:finite 1761:Finite 208:scale. 4462:Ponor 4398:Ponor 4143:white 4139:black 4092:. -- 3858:4,321 3157:here. 2890:Moo! 2746:WP:OR 2667:would 2395:? -- 2127:WP:OR 1573:empty 1294:from 1158:Pizza 917:WP:OR 710:with 189:This 84:Seek 4693:talk 4651:talk 4613:talk 4599:talk 4580:talk 4558:talk 4540:talk 4518:talk 4466:talk 4402:talk 4381:talk 4360:talk 4348:Here 4339:talk 4324:talk 4298:talk 4266:talk 4240:talk 4223:talk 4193:talk 4175:talk 4155:talk 4147:Grey 4141:and 4117:talk 4098:talk 3972:talk 3958:talk 3940:and 3915:talk 3837:talk 3736:talk 3700:are 3683:talk 3669:talk 3647:talk 3561:talk 3532:talk 3505:talk 3460:talk 3415:talk 3367:talk 3344:talk 3330:talk 3308:talk 3290:talk 3269:and 3238:talk 3209:talk 3187:talk 3166:talk 3144:talk 3122:talk 3090:talk 3071:CMYK 3026:talk 3004:talk 2983:Zero 2968:talk 2954:talk 2921:talk 2896:talk 2875:talk 2848:talk 2829:talk 2808:talk 2766:talk 2752:and 2732:talk 2714:talk 2680:talk 2657:talk 2639:talk 2624:talk 2609:talk 2594:talk 2569:talk 2554:talk 2530:talk 2511:talk 2492:talk 2478:talk 2455:talk 2441:talk 2428:Per 2401:talk 2383:talk 2368:talk 2351:talk 2311:talk 2294:talk 2285:much 2270:talk 2250:talk 2202:talk 2187:talk 2183:JonH 2168:talk 2153:talk 2120:this 2087:talk 2036:talk 1999:talk 1985:talk 1899:talk 1871:talk 1848:talk 1833:talk 1810:talk 1769:and 1674:ISBN 1640:talk 1626:talk 1606:talk 1596:and 1552:talk 1533:name 1521:talk 1500:talk 1492:talk 1472:talk 1451:talk 1363:talk 1340:talk 1322:talk 1308:talk 1286:has 1258:The 1249:talk 1233:talk 1210:talk 1119:talk 1104:talk 1091:Done 1075:talk 1058:talk 1042:talk 1026:talk 1011:talk 991:talk 976:talk 961:talk 944:talk 936:that 925:talk 906:talk 890:talk 863:talk 847:talk 824:talk 803:talk 784:talk 755:talk 740:talk 724:talk 697:talk 656:talk 640:talk 625:talk 610:talk 595:talk 580:talk 550:talk 518:talk 498:talk 469:talk 455:talk 428:talk 73:and 4631:(t) 4452:(t) 3929:(t) 3881:(t) 3805:(t) 3772:or 3518:(t) 3484:(t) 3443:(t) 3234:JBL 3104:(t) 3067:RGB 3042:(t) 2419:(t) 1993:Ok 1822:and 1443:TMI 1302:". 1298:to 919:). 714:or 691:.-- 687:or 565:or 442:is. 389:??? 303:Top 4707:: 4695:) 4653:) 4615:) 4601:) 4582:) 4560:) 4542:) 4520:) 4468:) 4404:) 4383:) 4362:) 4341:) 4326:) 4300:) 4292:. 4268:) 4242:) 4225:) 4195:) 4177:) 4157:) 4119:) 4100:) 4071:∪ 4047:∅ 4026:, 3974:) 3960:) 3917:) 3855:− 3839:) 3755:− 3738:) 3685:) 3671:) 3649:) 3612:∈ 3601:, 3584:∈ 3563:) 3534:) 3507:) 3462:) 3417:) 3398:. 3395:: 3391:, 3387:, 3383:, 3369:) 3346:) 3332:) 3310:) 3292:) 3240:) 3211:) 3189:) 3168:) 3146:) 3136:is 3124:) 3092:) 3028:) 3006:) 2970:) 2956:) 2927:) 2923:• 2898:) 2877:) 2850:) 2831:) 2810:) 2768:) 2734:) 2716:) 2682:) 2674:-- 2659:) 2641:) 2633:-- 2626:) 2611:) 2596:) 2571:) 2556:) 2532:) 2522:we 2513:) 2494:) 2480:) 2457:) 2443:) 2403:) 2385:) 2377:-- 2370:) 2353:) 2340:a 2313:) 2296:) 2272:) 2252:) 2204:) 2189:) 2170:) 2155:) 2089:) 2038:) 2013:RS 2001:) 1987:) 1901:) 1893:. 1873:) 1850:) 1835:) 1812:) 1804:. 1747:. 1741:. 1723:. 1717:. 1702:ø. 1699:. 1695:. 1672:. 1642:) 1628:) 1608:) 1554:) 1523:) 1502:) 1474:) 1466:. 1453:) 1445:? 1429:∅ 1365:) 1342:) 1324:) 1310:) 1251:) 1235:) 1212:) 1175:) 1172:c̄ 1161:03 1121:) 1106:) 1077:) 1060:) 1044:) 1028:) 1013:) 993:) 978:) 963:) 946:) 927:) 908:) 892:) 865:) 849:) 826:) 805:) 786:) 757:) 742:) 726:) 699:) 658:) 642:) 627:) 612:) 597:) 582:) 574:. 552:) 520:) 500:) 471:) 457:) 430:) 146:: 138:, 134:, 130:, 54:; 4691:( 4649:( 4611:( 4597:( 4578:( 4556:( 4538:( 4516:( 4464:( 4447:– 4438:n 4434:n 4400:( 4379:( 4358:( 4337:( 4322:( 4296:( 4264:( 4238:( 4221:( 4191:( 4173:( 4153:( 4133:@ 4115:( 4096:( 4080:} 4077:x 4074:{ 4068:x 4065:= 4062:) 4059:x 4056:( 4053:S 4050:, 4044:= 4041:0 4030:. 4016:⁠ 4003:N 3994:⁠ 3970:( 3956:( 3913:( 3902:. 3876:– 3870:⁠ 3847:⁠ 3835:( 3794:⁠ 3782:3 3774:⁠ 3770:⁠ 3758:1 3747:⁠ 3734:( 3681:( 3667:( 3645:( 3639:1 3635:2 3616:N 3609:1 3588:N 3581:0 3559:( 3530:( 3503:( 3458:( 3413:( 3365:( 3342:( 3328:( 3306:( 3288:( 3249:) 3245:( 3236:( 3207:( 3195:@ 3185:( 3164:( 3152:@ 3142:( 3120:( 3111:@ 3088:( 3049:@ 3024:( 3002:( 2976:@ 2966:( 2952:( 2919:( 2894:( 2873:( 2846:( 2837:@ 2827:( 2806:( 2794:@ 2764:( 2730:( 2712:( 2678:( 2655:( 2637:( 2622:( 2607:( 2592:( 2567:( 2552:( 2528:( 2509:( 2490:( 2476:( 2453:( 2439:( 2414:– 2399:( 2381:( 2366:( 2349:( 2309:( 2292:( 2268:( 2248:( 2227:N 2200:( 2185:( 2166:( 2151:( 2115:. 2095:@ 2085:( 2066:( 2034:( 1997:( 1983:( 1897:( 1869:( 1846:( 1831:( 1808:( 1795:. 1751:. 1727:. 1681:. 1638:( 1624:( 1604:( 1575:. 1550:( 1519:( 1498:( 1490:( 1470:( 1449:( 1408:N 1386:N 1361:( 1338:( 1320:( 1306:( 1300:S 1296:n 1288:n 1284:S 1280:n 1276:n 1268:n 1264:n 1247:( 1231:( 1208:( 1167:d 1164:( 1117:( 1102:( 1073:( 1056:( 1040:( 1024:( 1009:( 989:( 974:( 959:( 942:( 923:( 904:( 888:( 861:( 845:( 822:( 801:( 782:( 768:☎ 753:( 738:( 722:( 695:( 673:☎ 654:( 638:( 623:( 608:( 593:( 578:( 548:( 535:☎ 516:( 496:( 467:( 453:( 426:( 397:. 315:. 214:: 140:4 136:3 132:2 128:1 125:: 58:.

Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.

Index