Talk:Natural number/Archive 4

Edit of Peano axioms
I did suggest a modified version of this section and humbly ask for constructive review and improvement. I also beg for not simply reverting it because of my notorious poor writing.

My intention is to shift this unproductive zero-or-one discussion to the background and focus this article more on the minimal properties of naturals. Purgy (talk) 10:09, 28 February 2015 (UTC)
 * Your version consists essentially in renaming 0 as distinct natural number. This terminology is unsourced (see WP:RS) and confusing for most readers. Before such a fundamental change, a consensus must be reached here. Therefore, per WP:BRD, I'll revert your edit, waiting for a consensus. For having a chance of getting a consensus in favor of your edit, I suggest you to provide reliable sources and convincing arguments for such a change of terminology. D.Lazard (talk) 10:45, 28 February 2015 (UTC)

In the few examples where mathematicians do not agree on definitions ("natural numbers" is one, "ring" is another) it is important not to sweep the distinction under the rug. Rick Norwood (talk) 12:24, 28 February 2015 (UTC)


 * Agree both with the reversion of the changes to Peano axioms. The extra wordiness added nothing and only confused things as people would wonder whether the extra words meant anything - which they don't. Also agree with Rick Norwood as far as Wikipedia is concerned, it is policy that articles should show the major points of view on a subject and not just pick one like a textbook might do. Dmcq (talk) 16:15, 28 February 2015 (UTC)

@ D.Lazard: I assume, pointing to Hilbert's remark about beer mugs will not convince you of the advantageous use of "distinct element" instead of 0 or 1, so I prepared the version below to discuss about. It should be according to your measures a less fundamental change.

@ Rick Norwood: I strongly oppose to the opinion that I would sweep distinctions under a rug, on the contrary. There is of course a disagreement in mathematical nomenclature, if rings have to contain a unit or not, and really there are fundamental differences in the corresponding objects. There is however no disagreement on the rigorous notion of natural numbers as generated by the Peano axioms, regardless if its "first" element is denoted by the symbol 0 or 1 (beer mugs!). The distinction of having an additive unity -generally denoted with 0- is not founded in the property of being a natural number, but only -in a minimal sense- by these natural numbers bearing an additive monoid structure, where the 0 and the successor map happily coincide with the unity and (+1). This is, however, on top of the natural numbers and not intrinsically part of them. Imho, this mentioned disagreement pertains only to the use of the symbol $$\mathbb N$$ in denoting the positive or the non-negative integers and not to the definition of the notion of natural numbers. A similar severe disagreement arises perhaps from the use of $$\mathcal N$$ for the same purpose. I'm deeply convinced that all professional mathematicians agree on the rigorous concept of natural numbers. Whenever it is of any importance to distinguish between the two notations, some higher concept is involved.

@ Dmcq: may I assure you that I really tried to give a meaning to every part of my added wordiness. Could you perhaps, please, help me to the meaningless ones?

____________________________________________________________________________

Peano axioms
Many properties of the natural numbers can be derived from the Peano axioms.


 * Axiom One: 0 is a natural number.
 * Axiom Two: Every natural number has a successor, which is a natural number.
 * Axiom Three: 0 is not the successor of any natural number.
 * Axiom Four: If the successor of x equals the successor of y, then x equals y for all natural numbers x and y.
 * Axiom Five (the Axiom of Induction):  Any statement, which is [true for the distinct element], and for which [the truth of that statement for any given number implies its truth for the successor of that number], is true for all natural numbers (the brackets should ease to identify the two antecedents of this axiom).

These are not the original axioms published by Peano, but are named in his honor. The original form of the Peano axioms denotes this distinct element "0" with the symbol "1". This choice is of relevance only when adding further structures (additive monoid) to this minimal concept of natural numbers, thereby requiring specific properties of 0 or 1 to conform to commmon arithmetic: 0 as additive unity and expressing the successor of x as x + 1. Replacing Axiom Five, which quantifies (any!) over propositions, by an axiom schema, one obtains a (weaker) first-order theory often called Peano Arithmetic. ______________________________________________________________________________

Closing remark: I humbly ask for some constructive comments. Purgy (talk) 10:40, 1 March 2015 (UTC)


 * I like your change to Axiom Two. Your change to Axiom Four has a misplaced quantifier.  At the end of the sentence, it suggests that all natural numbers are equal.  It would be acceptable (but not strictly necessary) at the beginning of the sentence. Your restatement of Axiom Five seems to refer back to your earlier version of the first four axioms, and does not make sense with the current version of the first four axioms.


 * In the following paragraph, sentence two is confusing, as is the last sentence. It is not clear if "(any!)" is intended as part of the sentence or as a comment.  In either case, I do not understand what it refers to.


 * Rick Norwood (talk) 13:14, 1 March 2015 (UTC)

Peano axioms (revised)

 * Axiom One: 0 is a natural number.
 * Axiom Two: Every natural number has a successor, which is a natural number.
 * Axiom Three: 0 is not the successor of any natural number.
 * Axiom Four: If for all natural numbers x and y the successor of x equals the successor of y, then x equals y.
 * Axiom Five (the Axiom of Induction):  Any statement, which is [true for the first element 0], and for which [the truth of that statement for any given number implies its truth for the successor of that number], is true for all natural numbers (the brackets should ease to identify the two antecedents of this axiom).

These are not the original axioms published by Peano, but are named in his honor.

The original form of the Peano axioms employs the symbol "1" to denote the first (distinct) element of the natural numbers instead of the symbol "0" used above. This choice is of any relevance only when the meaning of the symbols "0" and "1" is extended beyond the requirements of the Peano axioms, to make those conform to commmon arithmetic: "0" as additive unity, and expressing the successor of x as "x + 1". To achieve arithmetical behaviour of natural numbers at least two additional monoid structures (addition and multiplication), not granted by the Peano axioms, must be added to the minimal concept. Expansion of the additive monoid to a group by adjoining additive inverses, results in the concept of the ring of integers and shows the method for further generalisations.

Axiom Five is not a first-order logical expression, because it all-quantifies over statements("any statement"). Replacing this single axiom by an axiom schema of countably many first-order logic expressions, one obtains a (weaker) first-order theory of the natural numbers, often called Peano Arithmetic. This does not affect the properties of natural numbers, but only the set of theorems which are deriveable in the respective theory. ______________________________________________________________________________


 * Please see above again:
 * - Since the quantifier is contained in the "then"-part of an if-expression its scope does not pertain to "all naturals", but these are the quirks of non-formal languages. Nevertheless, I edited it to the front. I'm well aware of the habit of omitting all-quantifications but for axioms especially, I do like to have them.
 * - I revised the fifth axiom.
 * - I modified the 2. sentence, but I am not sure if you did not refer to the third.
 * - I wrote more on the topic of the third sentence in my reply of "not sweeping under the rug", and tried to be more explicit in a reformulation, however at the expense of wordcount. Omitting the sentence on the integers would reduce it.
 * - I spent a few more words to make the last sentence more accessible, since in my first attempt I tried hard to keep the wordcount low.


 * Thanks for the suggestions. Purgy (talk) 10:17, 2 March 2015 (UTC)

The quantifier in Axiom Four still does work. Now it says that all natural numbers have the same successor. To quantify Axiom Four, you need to say "For all natural numbers x and y, if the successor of x equals the successor of y, then x equals y.

In Axiom 5, the brackets are confusing, not helpful. Axiom Five should be as simple as possible, for example: "Any statement that is true for 0, and whose truth for x implies its truth for x+1, is true for all natural numbers.

In the later paragraph, the second sentence does not need the word "distinct". We can say that x and y are distinct numbers, meaning x does not equal y, but to say that one number is "distinct" is meaningless. Distinct from what?

The following sentence is long and misses the point. The numbers 0 and 1 are, respectively, the additive identity and the multiplicative identity. The axioms of the natural numbers can start with either.

A discussion of the properties of a monoid does not belong here, nor does the extension of the natural numbers to the integers belong here. Both may be appropriate elsewhere in this article, but the topic of this section is axiomatic definitions of the natural numbers.

Next, you have written:

"Axiom Five is not a first-order logical expression, because it all-quantifies over statements("any statement"). Replacing this single axiom by an axiom schema of countably many first-order logic expressions, one obtains a (weaker) first-order theory of the natural numbers, often called Peano Arithmetic. This does not affect the properties of natural numbers, but only the set of theorems which are deriveable in the respective theory."

The difference between first-order logic and second-order logic may have a place in the article Peano Axioms, but not in this article. The last sentence does not make sense. What is a theorem about the natural numbers if not a property of the natural numbers? The current section says it better.

Rick Norwood (talk) 18:25, 2 March 2015 (UTC)
 * Rick, you have been showing remarkable patience. Frankly, User:Purgy Purgatorio's comment to the effect that "This does not affect the properties of natural numbers, but only the set of theorems which are deriveable in the respective theory" disqualifies that user from editing this article as they clearly don't have the necessary background in logic. Tkuvho (talk) 18:31, 2 March 2015 (UTC)


 * Just for reasons of politeness wrt to the wordcount of the answers:


 * @Rick Norwood
 * - I am not sufficiently a linguist to discuss the scope of quantifiers in if-statements of natural language, nevertheless, I keep up my POV.
 * - One could ask for a majority to decide on any improvement of these bracket, I certainly do not.
 * - I experience the beer mug mentioned in the first axiom as a "distinct" element for obvious reasons. Who am I to doubt your "distinct" opinion on this?
 * - Imho, the result of strictly applying Peano's axioms only does not know about any operations and so also not of their unities, and as said already too often, axiom 1 fixes a "Hilbertian chair, table or beer mug" to start the natural numbers with it, and not an additive or multiplicative unity.
 * - I'd rather believe that this abstract considerations of introducing additional structures, like monoids, belong to this axiomatic section rather than to the many other places it is now in the article. The set theoretical part of definition of naturals corresponding to Peano's axioms does neither employ the "0" nor the "1". Additional axioms for the operations are always required.
 * - Mentioning the variant of a FOL theory is contained in the current version of this article, giving a reason in few words seems appropriate to me.
 * - For the meaning of the last sentence see below.


 * @ Tkuvho


 * It's difficult for me to decide on a polite answer to your affronting disqualifaction, but I am convinced that this was intended by your comment. I am quite certain that you, with your highly qualified necessary background in logic, were able to express the fact to which I tried to allude in my imbecill way in an adequate manner to which Rick Norwood perhaps is not apt to.


 * Save again? Purgy (talk) 08:21, 3 March 2015 (UTC)

Where do we start?
I think the distinction between natural number and counting number should be made plain. Kids come here looking for answers, when they get marked wrong on a math test. The first paragraph, at least, should be accessible to a 10-year-old.

Kids and their school teachers - whether in primary, elementary, "grade" or "middle" school need to know the definitions of counting number, whole number, natural number, and integer. If there are variations, then we need to point those out. It's not fair to marked wrong if your teacher uses one definition for a trapezoid while the high-stakes standardized test uses another. We all need to know how the basic definitions and any variants.


 * 1) Positive numbers which are multiples of 1 are called counting numbers, right? (Unless you count from zero: "There are zero defects in this product.")
 * 2) Zero, along with the positive integers, comprise the whole numbers - or is it the natural number?
 * 3) The integers might be the easiest for teachers define: ..., -3, -2, -1, 0, +1, +2, +3, ... because no matter whether you start the natural numbers at zero or one, all the opposites of the positive counting numbers are included. (It's only hard for students.)

Can we make a chart or table? --Uncle Ed (talk) 15:31, 21 February 2015 (UTC)

>>No, sorry, your definition of "counting number" doesn't work. "Multiples of 1" assumes 1 is multiplied by something. That "something" is a counting number. So your definition boils down to "Positive numbers which are 1 multiplied by a counting number are counting numbers, a circular definition. The basics of mathematics are much harder to get right than most people suppose.

>>Are the positive integers together with zero the "whole numbers" or the "natural numbers"? Some say one, some say the other, and nobody has the authority to settle the matter.

>>You're right, the integers are easiest. But the way elementary education is run in the US, by thousands of local school boards making their own decisions and rejecting any attempt at coordination, the situation is impossible to fix. We should be glad all the other countries get it right, and will be able to keep civilization running after our own students can no longer do basic arithmetic. (I'm told we're rapidly losing ground in reading and writing as well, not to mention speaking, which isn't taught in most schools. Modern children text, and in many cases are unable to communicate verbally with their peers.)

>>Rod Pierce is a good resource, and he gets it right. Thanks for adding him. Rick Norwood (talk) 23:58, 21 February 2015 (UTC)

In my opinion it requires a high(?) mathematical scrutiny to really focus on the axiomatic properties of "the natural numbers" only, and not having their extensions in mind. There is up to my knowledge no primary (undergraduate?) curriculum dealing precisely with these finesses. However, there are teachers out there who rely in testing on pupils' ability to cling litterally to exactly their own, specific nomenclature. Since the academic math world does not agree (for economic reasons) on one single term for the result of axiomatically generating the natural numbers (a "magma"?), but only agrees on the concept of unity belonging to algebraic structures of monoids and up, the discrimination between whole, counting, and whatmore numbers of this kind is of marginal importance, imho.

Rod Pierce might get it right, but it should not be necessary to look him up for this, because of zelotic believe in just one nomenclature in teaching naturals. Rod Pierce seems to get this right also, but there are too many teachers insisting on their hobby horses, considering them to be relevant math.

Educating with these negligibilities is nitpicking but not paradigmatic for math's precision. Purgy (talk) 09:58, 22 February 2015 (UTC)
 * Hey everyone, this isn't Childcraft. Wikipedia is a serious encyclopedia &mdash; think Brittanica.  It is not our mission to cater to ten-year-olds.  If kids are trying to learn from a grownup encyclopedia, I think that's great, seriously.  But one of the most important lessons they will, and should, learn from that, is that sometimes things are more complicated than they teach you in fifth grade. --Trovatore (talk) 19:32, 22 February 2015 (UTC)
 * I agree on that. Additionally how certain sets of numbers are called is dependent on where you go to school and I can imagine the nomenclature is even in the English speaking countries not homgeneous. Citogenitor [talk needed] 12:14, 25 February 2015 (UTC)
 * Sorry, I've removed Rod Pierce as a reference. The website is of dubious mathematical rigor.174.3.125.23 (talk) 04:05, 23 February 2015 (UTC)

There is relatively universal standard terminology for the natural numbers: the non-negative integers include zero, the positive integers do not. All other terms I know do not have their relation to zero built in in a similarly obvious way and therefore are used sometimes in one way, sometimes in the other, but most often ambiguously. You can still be precise when using them: By saying explicitly whether you want to include zero or not – each time or once in the terminology section of a book or paper.

Apparently some educators believe that the term integer, qualified by non-negative or positive should not be introduced at the time of introduction of the natural numbers, which is long before the children officially get to see negative numbers. This makes some sense, though in my opinion mysterious terminology that foreshadows later developments isn't necessarily a bad thing.

What I don't agree with is the obsession with defining the natural numbers so precisely at such an early stage. Or at all, for that matter. The world is full of ambiguities of this kind. Penguins can't fly. Are they still birds? Kangaroos have no placenta but give birth (and breast milk) to living young. Are they mammals? Platypuses lay eggs and breast-feed. Are they mammals? Is a truck a car? Is our Sun a star? Are humans animals? Are tomatos a vegetable? Children grow up with these ambiguities. Some of them have been resolved by some kind of more or less general convention, but even then general usage in everyday speech doesn't necessarily agree.

There is no real problem involved with saying that the natural numbers, or counting numbers, are the ones we use for counting. In a way you can still count your elephants even if you haven't got any, so it makes *some* sense to include a special number zero for nothing at all. But in a way it doesn't make sense because when you are counting zero elephants you are not actually doing anything. Sometimes we will include zero because it makes sense, and sometimes not. Before assuming that zero is included, always think about what that would mean and if it makes sense.

Creative uses of zero to say ordinary things in a funny way is a great way to get a little child interested in mathematics. ("Did you hear that? I think there have been zero accidents over there!") Since children love paradoxes, there is no need to define them away.

If you make a general statement about the natural numbers, first think about the 'normal' ones. And then, if there is time left, think about whether it's also true in the special case of zero.

Since this is clearly how the natural numbers should be introduced, it is not surprising that some regions or at least individual teachers actually do that and the term counting numbers is used there as a full synonym for natural numbers including the ambiguity. With appropriate references we might be able to make a table of precise meanings of 'natural number' and 'counting number' in schools all over the (English-speaking) world. But I have looked for such references in the past and not been able to find any. Not even for individual countries or states. If someone knows how to look this up, that would be a great addition to the article.

But it's certainly not Wikipedia's job to give the impression that a specific choice of terminology is universal when it isn't and contradicts what children in some schools are learning. Hans Adler 08:19, 23 February 2015 (UTC)

>>Children are not the only ones who need a clear explanation of the two ways in which the phrase "natural numbers" is used. And, while Wikipedia is not primarily for children, neither is it primarily for experts. A clear explanation is a good thing. Everything that Rod Pierce says is mathematically correct. He knows what he is talking about. Why remove him?

>>Mathematics and science depends on accurate definitions. To answer Hans Adler's questions: Birds are animals with feathers. Flying has nothing to do with it. Mammals are animals that produce milk to nourish their young. Kangaroos, platypuses, and humans are mammals. The Sun is a star. The words "car" and "truck" are not scientific terms, but there are legal definitions of those terms, and the legal definitions are important. Praise of sloppy thinking does not impress me.

>>There is absolutely no use in arguing about what the definition of "natural number" should be. There are two definitions, and no logical way of choosing one over the other, so anyone interested in the subject needs to be informed of that fact. This article does that. Nobody in their right mind will ask a student if 0 is a natural number on a test. It's time for us to move on.

Rick Norwood (talk) 16:33, 23 February 2015 (UTC)
 * Your last paragraph above is fine. By far the best approach is just to state that there are two conventions, one includes zero and one does not, and as you say, move on.  That's all we have to say.  We don't have to go into details about how algebraists do one thing and analysts do another, unless the Moon is full or they're in grade school.
 * I don't have a huge problem with the Rod Pierce link (as an external link), but I am a little concerned that he categorically excludes zero from "counting numbers". I, for one, can easily count to zero.  In fact I do it all the time, without even thinking about it, or indeed doing anything at all.  But, while I think that's a flaw, I would not oppose restoring the link, if others want it. --Trovatore (talk) 18:40, 23 February 2015 (UTC)


 * I don't have a huge problem with the Rod Pierce link either, however, I'm in doubt if Wikipedia should strive to include sources of this specific kind (targeted clientel, professional deepness). I'm more with Hans Adler and Trovatore that this is not Childcraft. Rick Norwood's answers to the mentioned biological vaguenesses is, imho, at the heart of this problem. Some cannot live with negligibilities being undecided and others bath in undecidability as a matter in principle (participants of this discussion excluded).


 * Therefore I also support the opinion of prefering the use of non-negative integers and positive integers even for the uninitiated, because of their curiosity awakening potential.


 * I did not research the number of ways to axiomatically introduce naturals, but I confess that my most formative started with There exists a destinct element. Nevertheless, I consider the differnt opinions about zero-included in various names of variants of the naturals in no way noteworthy in a sense, that this "ambiguity" were necessarily to be documented by citations. Imho, interpreting this distinct element as one or zero or still something different is not core part of an encyclopedic article about natural numbers.


 * This is not to say that the deep intrincacies of the concept of zero are not worth to mention. Purgy (talk) 08:26, 24 February 2015 (UTC)

Thanks, Hans and Rick. I'm hoping that even if we can't provide a one-size fits-all definition for natural and counting numbers (that satisfies college grads as well as schoolchildren), at least we could make some progress toward listing the first few members of the various sets of numbers.

I'm not actually proposing a definition of natural or counting numbers, but asking for a clarification of the various existing definitions.


 * positive integers: {1, 2, 3, ...}
 * non-negative integers: {0, 1, 2, 3 ...}

Which of the above is the definition for natural or counting numbers? And in case there is no universal agreement, what are the major variations?

Since I teach math in New York, I'm mostly concerned about the Regents tests giving each spring to kids in NYC. At some point my students will also be taking the College Boards given by ETS.

I hope no one will object if I create a table (somewhere at Wikipedia) giving the simplest possible definitions, any major variations, as well as links to the comprehensive articles provided by math experts such as yourselves. --Uncle Ed (talk) 17:07, 3 March 2015 (UTC)


 * I don't think such a table is necessary, since exactly what you ask for is already in the lead. Also, I trust the people who write the Regents tests are smart enough not to ask for a definition of the natural numbers. Rick Norwood (talk) 15:47, 4 March 2015 (UTC)
 * Does he mean a table for a separate article?174.3.125.23 (talk) 19:10, 4 March 2015 (UTC)

Shooed away
This article seem to be protected not only from vandalism and deterioration, but from any change and improvement.

Many efforts, only to a small part from my side, have been simply reverted for being poorly written. There was no constructive, accepting improvement. If there was a discussion, then only to string the contributor around, rebuffing all of their suggestions, even in modified, adjusted form.

This protection seems to be exerted not only from one single person, but from a hierarchical team already, the only persued target of which seem to keep any changes off this article, which itself might be not so bad, but really could use some punctual improvement. This improvement is prohibited by the actions of this (informal?) team.

You mayy check this process in all the currently 6 sections of the talk page, going to be automatically archived not earlier than 25.03.2015. I do not consider this to be the best development of Wikipedia. Purgy (talk) 08:48, 3 March 2015 (UTC)
 * Purgy, you seem forget that an axiomatic construction has no value by itself (one can easily define infinitely many axiomatic systems which have absolutely no interest). Its value lies in allowing rigorous reasoning (that is secure truth of the result) on abstractions that help to model the real world (here numbers). Thus you are wrong by saying that "axiom 1 fixes a "Hilbertian chair, table or beer mug" to start the natural numbers". The truth is that it specifies a distinguished natural number. To decide if this distinguished number is 0 or 1 is partly a question of taste, but the main reason is to have a simpler construction of all mathematics above this starting point. In any case, changing 0 into 1 in the first axiom implies to change all subsequent definitions. For example, the definition of the addition must be changed from $a + 0 = a and a + s(b) = s(a + b)$ into $a + 1 = s(a) and a + s(b) = s(a + b)$. The second definition is not obtained from the first one by changing 0 into 1, and the first one is simpler, as the successor function is not involved in the start point of the recursion. The difference is small in this case, but for more involved definitions, such as that of Euclidean division of natural numbers, the difference becomes dramatic (how to define a zero remainder if you do not have zero?).
 * Above considerations are partly WP:OR, and my opinion is not shared by all mathematicians, even if I believe that most agree. Thus these considerations cannot be included in this article. However, the article must be acceptable by every mathematician. As the axiomatization of mathematics and the question "what is a number?" involve epistemological considerations on which there is not a strong consensus, every major change of this article (as is the one that you propose) requires a strong consensus for insuring a neutral point of view.
 * Therefore, before proposing a change to the article, you must clearly explain why the present version is not convenient and which issue of the article needs to be corrected. You wrote "I did suggest a modified version of this section and humbly ask for constructive review and improvement" at the beginning of this section. This has been reviewed, and there is a clear consensus against your proposition. On your side, you have not done your part of the job, which is to explain why you think that a modification is needed. D.Lazard (talk) 09:50, 3 March 2015 (UTC)


 * , as said, I think that for politeness reasons at least, an aswer to your efforts is appropriate.


 * - Please may I refer you to the meaning of me citing repeatedly beer mugs, which refutes on D. Hilbert level your opinion about "0" or "1" being fixed for their meaning of distinguished naturals (I notice that I used -possibly erroneously- the word "distinct" for your "distinguished").
 * - A main point of my efforts to improve on this article was to get away from the wrong assumption that the naturals -as defined by the Peano system- have anything to do already with arithmetic. Only when adding these mentioned further structures one has to decide about the tokens (numerals?) to employ to not contradict habitual use. The naturals -as Peano-naturals- do not care if "0", "1" or "beer mug" is their "first" element, as long as there is a successor. The hope that "1" is the successor of "0" is not to be based on Peano's, but just on ubiquitious habit. There is absolutely no arithmetic generated immediately by the Peano Axioms! The successor map just gives the setting to plant an operation "add" based on "repeated successing" by amending further axioms.
 * - Starting from Peano's axioms (in the neutral formulation of "distinguished" element) there are several ways to introduce arithmetic:


 * One could start with the operation add, implementing a semigroup without additive identy, simply taking the distinguished element as the element to "add" to another to get its successor. In a next step one could define "multiply" as "repeated addition", using the distinguished element as multiplicative unity, establishing a monoid for this operation. This would stimulate to denote the distinguished element by the token "1". In a next step one could suitably adjoin an additive identity and additive inverses for the naturals, this would bring forward the token "0" and the minus-sign to fit the common expectation, delivering the ring of integers.


 * Now take totally the same(!) set of axioms as before:


 * One could start with the operation add, implementing now a monoid (instead of a semigroup) with an additive identy, simply taking the successor of the distinguished element as the element to "add" to another to get its successor. In a next step one could define "multiply" as "repeated addition", using the successor of the distinguished element as multiplicative unity, establishing a monoid for this operation too. This would stimulate to denote the distinguished element of the naturals by the token "0" and its successor by "1". In a next step one could suitably adjoin additive inverses for the naturals, the distinguished element being its own inverse, bringing forward the the minus-sign to fit the common expectation, delivering the totally same ring of integers.


 * Sorry for boring you with almost identical paragraphs, but please note, that in both paths from the naturals to the integers 'one and only one' concept of naturals as given by the Peano Axioms was employed, with only marginal differences in the necessary steps.
 * - I do not consider it possible to discuss divisibility in the realm of naturals as founded by Peano's because of lack of arithmetic in Peano's, and in my opinion the integers with their ring structure, which requires additional "axioms", are the appropriate environment for Euclid's.
 * - Requesting an explanation before allowing any edit in Wikipedia is imho in strict contradiction to boldly editing and improving. To my perception I've implemented all suggestion which were not factually wrong, I even conceded to formulation details, but did not experience any constructive improvement, rather impolite and offensive rebuff, instead. All this has been answered by blatant offensively discrediting my qualification.
 * - I fully take the consensus against my opinion as given, but I do not consider it being achieved on sufficiently neutral and strict grounds, not to talk about appropriate scientific openess (strict personal POV).


 * Please, assume that I did not perceive any offense from your side and that I hope you did not from my side. Purgy (talk) 08:43, 5 March 2015 (UTC)

linking to 19th century
I've reworded the phrase that includes "19th century". I believe it should be linked. Objections?174.3.125.23 (talk) 23:58, 4 March 2015 (UTC)
 * My intuition is that it should probably not be linked. I don't think a passage on mathematical developments in the 19th century is a terribly natural place for a reader to say, "oh, that reminds me, I want to find out more about the 19th century in general".
 * Anyway, that's just my impression, since you asked. If you feel like linking 19th century, I'm not going to revert you.  Others may feel differently. --Trovatore (talk) 05:10, 5 March 2015 (UTC)
 * Ideally, to improve such a powerful crosslinkable encyclopedia, proper nouns should be linked. I'll go ahead and link the century.174.3.125.23 (talk) 02:41, 6 March 2015 (UTC)
 * Oh, no, sorry, that's completely the wrong criterion. See WP:OVERLINKING.  Absolutely no way all proper nouns should be linked.  Link only when there's a reasonable probability a typical reader would want to follow the link.
 * I promised not to revert you so I won't, but I invite you to undo this edit after reading the guidance I linked to. (Europe should not be linked either, and the two links right next to each other are especially problematic, because it looks like a single link to 19th century Europe.) --Trovatore (talk) 04:14, 6 March 2015 (UTC)

ClueBot NG hides ID
I do not critisize the revert which has been done by the bot on the article page in removing the word "incorrectly", but I consider judging this to be beyond a bot's capability.

I wanted to have this checked and tried to get to the required ID but failed. This ID is truncated, possibly because of the lenght of an IPv6 address. I could not deal with the variant "on my talk page" and also looking for the edits of ClueBot NG did not make this ID available to me.

I apologize if I abused this page by posting this kind of trouble here. Please, let me know where it were appropriate, in case. Purgy (talk) 08:55, 5 March 2015 (UTC)
 * I agree. It seems like a purposed use of hiding someone's ip address with such a blatant contradiction of the definition.174.3.125.23 (talk) 02:44, 6 March 2015 (UTC)
 * Yes the revert seems good, being done by a bot does not. Paul August &#9742; 14:09, 6 March 2015 (UTC)

paragraph removed
I have removed the paragraph
 * The attempt by Frege mentioned above, as modified by Russell, where each natural number n is defined as the set of all sets with n elements has been modified to avoid paradoxes. This definition at first may appear circular, but can be made rigorous with care. Define 0 as  (clearly the set of all sets with zero elements) and define S(A) (for any set A) as {x ∪ {y} | x ∈ A ∧ y ∉ x} (see set-builder notation). Then 0 will be the set of all sets with zero elements, S(0) will be the set of all sets with one element,  will be the set of all sets with two elements, and so forth. The set of all natural numbers can be defined as the intersection of all sets containing 0 as an element and closed under S (that is, if the set contains an element n, it also contains S(n)). One could also define "finite" independently of the notion of "natural number", and then define natural numbers as equivalence classes of finite sets under the equivalence relation of equipollence. To avoid the paradoxes that occur in the usual systems of axiomatic set theory (because the collections (classes) involved are too large), we need to drop the axiom of separation); the resulting variant of set theory is called New Foundations. There are other attempts to reformulate set theory without the axiom of separation, and these variants have been shown to be consistent if New Foundations is consistent. Another approach is found in some systems of type theory.

In fact, except for the first sentence, it is not sourced. It mention the "set" {x ∪ {y} | x ∈ A ∧ y ∉ x} and talk about the "set of all sets with one element", which are clearly not sets, by Russel's paradox (does {S(0)} belongs to S(0)?). It asserts without sourcing that there are set theories such as New Foundations that may solve these paradoxes, but does not mention that the text of the paragraph is wrong in the most used set theory named ZF. Thus it is unsourced, mathematically wrong and does not have a neutral point of view. Therefore I have removed it. D.Lazard (talk) 21:09, 6 March 2015 (UTC)

For clarification's sake, ===Modern definitions=== does mention Frege, but probably in a historical context, not a mathematical context.174.3.125.23 (talk) 22:12, 6 March 2015 (UTC)

For the record that paragraph was added here by. Paul August &#9742; 22:51, 6 March 2015 (UTC)

"count from zero"
In the latest version, "count from zero" was removed. I understand it was rewritten to include the "historical" context behind the convention of including zero. Just thought I'd bring it up to explain that the link could be piped to include the deeper topic of including zero for "computer programmers".96.52.0.249 (talk) 10:51, 4 July 2015 (UTC)


 * Does my last linking the job for you in sufficient manner? Honestly, I did not expect an article on this, even if it misses the true origin. I'll look after it. Purgy (talk) 17:17, 4 July 2015 (UTC)

Debated sentences.
Purgy Purgatorio wants the following sentences in the lead and reverted my deletion of it. "This distinction is of no fundamental concern for the natural numbers as such, since their core construction is the unary operation successor. Including the number 0 just supplies an identity element for the (binary) operation of addition, which makes up together with the multiplication the usual arithmetic in the natural numbers, to be completed within the integers and the rational numbers, only." Please explain what "to be completed within the integers and the rational numbers, only," means.Rick Norwood (talk) 17:49, 4 July 2015 (UTC)


 * This last clause is intended to refer to the arithmetic, which cannot be completed within the naturals but requires the mentioned extensions. May I cordially invite you to find a more suitable formulation in elegance and succinctness?


 * It is important to me to state, that I did not revert anything, but reformulated a deleted content, putting an aggravated distinction in a more reasonable perspective, to conform to mentioned deficiencies. Purgy (talk) 15:36, 3 July 2015 (UTC)

Sorry for my error. I now understand your reformulation. Rick Norwood (talk) 12:58, 4 July 2015 (UTC)


 * No problem! My invitation is still sustained to find a better formulation for the intended fact. I'm not perfectly satisfied myself, but also not sufficiently eloquent in this. Purgy (talk) 17:23, 4 July 2015 (UTC)

Block LMAOBOX
The actions taken by the mentioned above id look to me as pure, intentional vandalism. Could some mod take take care of this? Purgy (talk) 05:43, 10 September 2015 (UTC)
 * The user was blocked at 00:23, five hours and twenty minutes before you posted. ;) Richard Harvey (talk) 07:28, 10 September 2015 (UTC)


 * , thanks for this information. I am sorry for being such a newb, not knowing where to look for appropriate info. Should I have interpreted the id appearing in red that this user is blocked, already? I know this color also from users whose home is an other wikipedia-language. Purgy (talk) 12:45, 10 September 2015 (UTC)


 * If the user name is a red link, it usually just means that they don't have a user page. No crime there. Even without a user page, many users have talk pages where you might see a notice that the user has been blocked (they're not always updated, though). If you go to "contribs" for that user, I think the block notice is guaranteed to show up there, near the top. On a user talk page, the left-side menu also has a link to "User contributions". Willondon (talk) 12:56, 10 September 2015 (UTC)


 * No apologies required for being a newb. 😊 We all started the same way. Richard Harvey (talk) 13:01, 10 September 2015 (UTC)

Thanks for the info and the nice words. :) Purgy (talk) 05:48, 11 September 2015 (UTC)

Semiprotected
The article has been semiprotected due to IP edit warring per a complaint at WP:AN3. If you disagree with the way this article defines a whole number, use an WP:RFC or one of the other methods of WP:Dispute resolution. EdJohnston (talk) 15:17, 4 October 2015 (UTC)

Distinction between whole numbers and natural numbers
Hello, it appears there is a lot of confusion between whole numbers and natural numbers, not just on the wikipedia page but elsewhere on the internet and in text books so I wanted to do my part to try to clear that up. There is one very small difference between the natural numbers and whole numbers. Natural numbers are defined as your counting numbers, i.e. 1, 2, 3, and so on. Whole numbers are all of the natural numbers with the inclusion of zero. So whole numbers have zero, natural numbers do not. I know I am not good at convincing people of anything, but those are the definitions so I wanted to do my part and try to improve the page on natural numbers with that small change, however, my edit keeps being undone. Just remember the beauty and simplicity of mathematics, if natural numbers and whole numbers were exactly the same, why would you give two different names to the same set? — Preceding unsigned comment added by 107.205.236.24 (talk) 02:54, 4 October 2015 (UTC)

Nevermind, I went back and read the archives. So many people are convinced that the wrong definition of Natural Numbers is the correct definition that I guess I can't help correct it. I hope someone with enough influence is able to correct the errors on these pages and able to correctly educate people on the difference of Natural Numbers and Whole Numbers. — Preceding unsigned comment added by 107.205.236.24 (talk) 03:33, 4 October 2015 (UTC)


 * I'm strongly convinced that in terms of Wikipedia you have no righteous claim of pointing to the "correct" definition of any notion in math. The current realm of textbooks and other notable sources makes it, also imo, necessary for Wikipedia to point to the different POVs in a sort of equidistant manner. Definitely, you violate this standard.


 * Some months ago I tried to take out some of the lurking controversy by emphasizing that neither 0 nor 1 are relevant to the notion of natural numbers, but only get meaning in some structure higher up. I do not believe that it might comfort you, that not even this was welcome.


 * Regarding your last edits, using the generic term "number" in some axioms for the naturals is imho absolutely inappropriate. I think it would show some cooperativeness if you reverted this yourself.


 * Please,try to get concordance on the talk pages for evidently controversial edits. Purgy (talk) 07:14, 4 October 2015 (UTC)


 * please read my comments on your talk page about your edits to both this article and the ordinal number article. -- The Anome (talk) 10:53, 4 October 2015 (UTC)

Sorry you're all wrong. Bugger off. — Preceding unsigned comment added by 107.205.236.24 (talk) 22:45, 4 October 2015 (UTC)

Anyone who writes their own book gets to choose the terminology that they use. Unfortunately, they don't all choose the same terminology. On Wikipedia, we try to describe the terminology as it is commonly used, rather than prescribing how it ought to be used. There are many different situations or contexts in which the term "natural number" is used. Three of them are childhood mathematics education, number theory, and set theory. In childhood education, they tend to use many terms: counting number, whole number, natural number, etc. I see these terms much less in more advanced contexts, where just "natural number" or $$\mathbb{N}$$ seems to be common. In number theory, they often take $$\mathbb{N} = \{1,2,3,\ldots\}$$, as this is convenient for them; in set theory, they often take $$\mathbb{N} = \{0,1,2,3,\ldots\}$$. There is nothing we can do about this seeming inconsistency between different authors, except to explain it as clearly as possible. One challenge is that each book tends to talk about its own conventions, but not about other possible conventions. So we have look carefully to find good sources. &mdash; Carl (CBM · talk) 12:46, 4 October 2015 (UTC)

Edit request
Okay, I've looked through the talk page archives and I realise this is a controversial topic. But anyway I wanted to give my point of view as someone viewing this article as it stands now.

In the first three paragraphs of the lead, the definition of "whole numbers" (as naturals with zero) appears three times. The second and third repetitions could be removed without losing anything. The second repetition is hijacking a paragraph: "Some authors begin the natural numbers with 0" was clearly intended to be the start of the paragraph, introducing the topic (a balanced look at the two meanings of "natural number") the paragraph talks about.

In the next paragraph, "The whole numbers are the basis from which many other number sets may be built by extension", "whole" should be replaced with "natural". The natural numbers are the topic of this article. "Whole numbers" is an informal name not in serious use by mathematicians. It is enough to mention once that "whole numbers" is a synonym, after which "natural numbers" should be used.

This sentence is unwieldy: Including the number 0 just supplies an identity element for the (binary) operation of addition, which makes up together with the multiplication the usual arithmetic in the natural numbers, to be completed within the integers and the rational numbers, only. Better: "Including the number 0 just supplies an identity element for the (binary) operation of addition, which together with multiplication makes up the usual arithmetic of the natural numbers." I removed the last clause, because the integers and rationals do not complete the set of natural numbers; they extend it, and this is already covered in the next paragraph.2.24.117.123 (talk) 21:28, 7 October 2015 (UTC)


 * This kind of jumbled intro is what happens when editors have serious differences of opinion. You might say that these are the battle ruins of former edit wars. Some of these statements were placed where they were to deal with very specific issues at the time and were not necessarily viewed with the overall picture in mind (I am myself not above blame in this). Overall I agree with your comments and would like to apply fixes, but there are a couple of issues that I would like some agreement on before doing anything serious. First of all, I would say that natural numbers and whole numbers are only sometimes synonyms. I don't think that I have seen anyone argue that the whole numbers, although not a term that is used technically, consist of anything other than zero and the positive integers (oh somebody will argue that zero is considered positive by some - I will just ignore this as it makes no difference to my statement.) The ambiguity lies with whether or not zero is considered a natural number. For one camp (zero is a natural number), whole number and natural number are synonyms, but for the other camp they are not. I don't see how I can implement your suggestion and still please both camps. In the sentence starting, "The whole numbers are the basis ...", I think the intent was to definitely include zero, so "whole number" unambiguously does this, but using "natural number" would require some doctoring. As to the run-on sentence - it definitely needs the period where you placed it. The intent of the clause, as I see it, is to indicate that having the additive identity makes the extension to the integers more natural (no pun intended) as well as the further extension to the rationals. This is alluding to the ring (and field) properties which the next paragraph doesn't really do. I do think that something along these lines should be mentioned here, but that clause just doesn't do it right. Any thoughts, anyone! Bill Cherowitzo  (talk) 22:39, 7 October 2015 (UTC)
 * Actually there does seem to be a contingent that treats "whole number" as synonymous to "integer" (positive, negative, or zero). It seems to be significant enough that we have to deal with it.
 * That said, as you correctly noted, "whole number" is not a term that's used much at all as a technical term in research mathematics, and this article is not primarily about terminology. So my preference is to say about it the absolute minimum we can get away with.  A single cleverly crafted sentence should be enough. --Trovatore (talk) 22:45, 7 October 2015 (UTC)
 * Here's my attempt. It certainly might be improved.  But I think it redimensions the term "whole number" to its correct level of importance (that is, very very small), and also deals with the "integer" meaning, which we probably have to.  If I recall correctly, there is also a contingent that treats "whole number" as meaning "positive integer", and I have not dealt with that one.  Not sure if we need to. --Trovatore (talk) 23:00, 7 October 2015 (UTC)
 * Looks good. Probably have to bold "whole number" as it redirects here. Maybe adding something like, "Being a non-technical term, definitions may vary." so that we are covered without having to specify all the variations. Bill Cherowitzo (talk) 23:14, 7 October 2015 (UTC)
 * Hmm, you're probably right on the bold. I'm not so enthusiastic about "definitions may vary", which seems awfully vague.
 * I would prefer to find out whether the "positive integer" meaning has enough currency to be worth mentioning, and either call it out explicitly or let it be. Then the sentence probably does need to be referenced, though not so heavily as the stuff I removed.  Anyone feel like trudging through the versions to see if there are citations worth resurrecting? --Trovatore (talk) 23:40, 7 October 2015 (UTC)

As a recent contributor to the mentioned "edit dissents" (it was not a "war" to me), I propose to ponder on an alternative formulation for "including an unresolved negation operation". I suggest "including an additive inverse (-n) for each natural number n (and zero, if it is not there already, as its own additive inverse)" or a similar variant. In the same ghist, instead of "including with the integers an unresolved division operation" I'd prefer "including a multiplicative inverse (1/n) for each integer number n" or a similar variant. Finally, instead of "Therefore, the natural numbers are..." I suggest to be more explicit, perhaps with "These chains of inclusions make the natural numbers ..."

I did not want to implement these changes by just being bold, and would do so only if sollicited. Purgy (talk) 10:19, 8 October 2015 (UTC)


 * Certainly your formulation is an improvement. In your last quote, I suggest "These extensions make the natural numbers ... ." Rick Norwood (talk) 11:35, 8 October 2015 (UTC)

In performing the offered edits I stumbled across "vectors and matrices" as extensions of natural numbers. As I showed in my edits of the extensions I fully agree to mentioning them up to the hyperreals. I do not know enough about "non-standard integers" to discuss them. However, I think that vector spaces to not really rely on "numbers", but rather on algebraic notions (fields and groups). BTW, I doubt the natural embedding of naturals in arbitrary vector spaces. In going along this road anything gets "number" (not being obviously false, but not being very useful).

I plead for removing the vector/matrix-sequence. Purgy (talk) 07:23, 9 October 2015 (UTC)


 * I agree. In fact, I would stop at the complex numbers, adding additional systems with the natural numbers as a subset in a separate section, if needed. Rick Norwood (talk) 12:12, 9 October 2015 (UTC)


 * I also agree. Hyperreal and non-standard reals are too WP:TECHNICAL for this lead, and almost nobody would consider matrices and vectors as numbers. Instead, one could add a sentence like Most of mathematics is built upon these various kinds of numbers, including modern geometry (ancient geometry was not built upon numbers; on the contrary, it was used to introduce them as measure of lengths). The sentence about inclusion of natural numbers in other sets of numbers seems misplaced in the lead. D.Lazard (talk) 14:18, 9 October 2015 (UTC)

Road to infinity
It appears to me that honoring the role of the naturals in accessing the infinity within the lede might be justified. The naturals being a nice index set did, afaik, allow for many a constructions already necessary to achieve the first relevant notions of continuity (reals), I also think that Cantor built his extensions on their existence, and Kronecker possibly estimitated their infinity as being top most (God made), not allowing for further Towers of Babylon. Perhaps his citation about the rest being human's work could also belong to the lede, not for its "inherent truth/falsity", but for depicting the importance of naturals in "ancient modern" math.

Just an other idea. Purgy (talk) 11:08, 11 October 2015 (UTC)

"Nominal numbers" paragraph
At this writing, the second paragraph says:
 * Another use of natural numbers is for what linguists call nominal numbers, such as the model number of a product, where the "natural number" is used only for naming (as distinct from a serial number where the order properties of the natural numbers distinguish later uses from earlier uses) and generally lacks any meaning of number as used in mathematics.

In my estimation, this sidetracks the flow of the article. The article is not about nominal numbers, which don't even have to look like numbers at all (your confirmation number for an airline reservation, for example, might be QDEC4TR, and they don't mean base 36 particularly).

I would prefer to move this paragraph out of the lead, or perhaps remove it entirely. If it were moved, any suggestions as to where it should be moved? --Trovatore (talk) 18:48, 8 October 2015 (UTC)


 * In the paragraph above that, introducing cardinal and ordinal numbers, I find "in common language" unclear. As opposed to what other sort of language? Are the words 'cardinal' and 'ordinal' common? Do we need any of the content mentioning nominal, cardinal and ordinal? It seems to refer to aspects of language rather than mathematics. I would support removing all of it entirely. I thought maybe a "See also" addition, but I couldn't find a good target. Willondon (talk) 20:16, 8 October 2015 (UTC)
 * I see your point. I think the reason to have them there is to follow up on the first sentence's "counting" and "ordering".  The reason for "common language" is to distinguish them from the set-theoretic notions of cardinal and ordinal, which we want to do for two reasons:  First, we'd like the early part of the article to be set-theory-agnostic, and second, because set-theoretic cardinals and ordinals might be infinite, whereas natural numbers are finite.
 * Removing the sentence entirely strikes me as plausible, but then we need something to replace it with. We don't want to start with a one-sentence paragraph.  If we remove some of this stuff that strikes you and me as deadwood, then we have room for some "live wood", and it would be nice to have some thoughts about what that would be.  In such a rich topic, there's bound to be something we can say, but I don't have any immediate proposals. --Trovatore (talk) 20:38, 8 October 2015 (UTC)
 * I do see a purpose in keeping the statement about nominal numbers, but it should certainly not be where it is. I don't see any place to put it in the current article, and it doesn't deserve a section of its own, so I would make it the last statement in the lead. I'll make the change, but will keep other options open. Bill Cherowitzo (talk) 02:00, 9 October 2015 (UTC)


 * Adding my (verbally) two cents:


 * - I think in linguistics the terms "ordinal/cardinal number" in their linguistic meaning are as common and frequently used as their homophones are used in set theory in the meaning which is generic there.


 * - In an increasing amount I get reserved against the "nominal numbers" being discussed under "natural numbers". As I mentioned already several times, the successor functionality is to me the core and defining property of the naturals. Neither "4711" nor "08/15" or their siblings have ever been designed to have unique successors for ever and ever (or have they?). This holds for PINs, TANs, ..., and whatever finite word over some alphabet may be construed, and usurpatorically being called "number", when just being some identifier, sometimes denoted by the same tokens as "true" numbers are. Especially, these are no "natural numbers", with a specific beginning and ongoing, unique successors allowing for math induction.


 * As a meager suggestion I'd deal with these "nominal numbers" in this here article only rudimentary as "labels, sometimes looking like representations of natural numbers". Purgy (talk) 07:06, 9 October 2015 (UTC)
 * I think the sentence is okay and that we should say something like that in this article rather than for example the Number article, I think Purgy's suggestion of saying 'label is a good way of describing them simply. Dmcq (talk) 13:17, 9 October 2015 (UTC)
 * OK, so you and Bill both seem to think it belongs here. Can you explain why?  I can see mentioning it at number, because that's a concept with fuzzy edges that at least arguably includes nominal numbers.  But this is the article about natural numbers, a much sharper-edged (and more specifically mathematical) notion.  What is the point of talking about nominal numbers in this article? --Trovatore (talk) 18:21, 9 October 2015 (UTC)
 * When faced with the question of what I think belongs or doesn't belong in an article, I often (and probably not often enough) ask myself who I think are the probable readers of the article and what are the questions that they are most likely to have which got them here. An article such as this is not going to appeal to someone with sophisticated mathematical abilities except in two cases that I can think of: 1) trying to recall some detail of the Peano construction that they have forgotten, or 2) settling the bar bet on whether or not 0 is a natural number. (It is somewhat sobering to consider the fact that Wikipedia may mainly be used to settle bar bets!) That leaves the bulk of the readership browsing the page out of curiosity or having some vague kind of question like, "what makes a natural number natural?" The purpose of the lead (I know I am preaching to the choir, but I will be making a point) is to summarize what is in the article so that such a reader can get a feel for whether or not their question might be answered. Also valuable are pointers to other places that might be more directly useful. Hatnotes are good for this but don't work too well when the distinctions are subtle, so sometimes you should build into the lead a glorified hatnote to help a reader find what they are after. I am viewing this statement in that light. For those readers with some vague notion of using natural numbers as labels, we should be directing them to the proper page (nominal numbers) because the chances are slim that they would ever find that page on their own. I think a similar, but slightly more expansive, statement should appear in the number page as well and for the same reason. Bill Cherowitzo (talk) 05:13, 10 October 2015 (UTC)
 * The lead is supposed to summarize what is in the article, yes. Not what is not in it.
 * The navigational point is one I might give some weight to, but I don't actually think it's very likely that someone looking for that concept would type "natural number" into the search box. It's too specific and technical a term, and not one that seems to me you would very likely come up with by accident for the "nominal" concept. --Trovatore (talk) 14:34, 10 October 2015 (UTC)
 * Don't forget the consequences of redirects. Typing in "whole number" (I think highly likely in this circumstance) would get you here. Of course you can't put into the lead everything not in the article nor can you anticipate all of the questions users are seeking answers for — but a little judicious prognostication on our part can help a large number of readers. Bill Cherowitzo  (talk) 18:18, 10 October 2015 (UTC)
 * Hmm, I don't think I agree that "whole number" is a very likely search term for this concept. In fact I can't think of one more likely than just plain "number".  Or perhaps "code" or "code number", but those don't really concern us here.
 * But in any case, I think we shouldn't go to too much trouble to guess how people might wind up at the wrong article. For really obvious things, sure, but this isn't one of those.  In fact I think it's probably a really unlikely thing to search for at all, under any search term.  Since it just really isn't what the article is about (we all seem to agree on that), my opinion is that we just shouldn't mention it at all. --Trovatore (talk) 18:32, 10 October 2015 (UTC)
 * I'll agree that my argument in this case is a bit tenuous, but I do think the general principle is sound. I think that "whole number" could be the search term choice for the crowd that wants to avoid any mention of fractions or decimals (and "number" would surely bring up those reprehensible topics). I do feel that the article is a little better with this minimal statement in (at least in the simpler language of my last edit), but I wouldn't be terribly upset if it goes. Bill Cherowitzo (talk) 18:52, 10 October 2015 (UTC)
 * Having looked at it again I think perhaps just the one line here is right. I was put off the idea of saying anything at the number article as it goes into lots of more advanced mathematical concepts a bit earlier than here. However it does also include a line in the intro about numbers being used as labels. Perhaps instead of saying label we should say identifier though. Dmcq (talk) 14:06, 10 October 2015 (UTC)


 * Returning to this question after reflecting a while on it, it appears to me quite advantageous to have some hint to a common flaw in understanding a notion, besides its formal definition, its informal description, its application, and perhaps some examples. This helps, especially the non-professionals, to better grasp the ghist (sorry, I "know" there is none ;) ) of the formalism. As far as I understand this, I sort of agree to Dmcq. However, I object to any sloppiness in formulating this sentence, since this would void its purpose and intention - furthering precise understanding. Better having no remark at all than an inprecise one. Purgy (talk) 07:40, 12 October 2015 (UTC)

While being at it, I simply boldly dropped my suggestion for this paragraph. Please, do not consider this as some inappropriate arrogation. Purgy (talk) 09:14, 10 October 2015 (UTC)

I want to deposit my objections to the edit of User:Wcherowi stating that "The natural numbers can, at times, appear ...", reasoned by "representation" being an "unknown" notion to some (many?) readers. I think that "representation" is an easy to grasp concept, even when missing some of the deeper intricacies, and one of the targets of this article, at least I see at the horizon, is to further an abstract understanding of the concept of (natural) numbers, and so I see mentioning "representation" at this very point quite essential, and not too demanding. Additionally, the remainders say: "The natural numbers ... appear as ... foregoing ...being a number in a mathematical sense."

On the other hand, I fully agree to exchanging "normal" to a simple "a", perhaps "usual" might be a (weasely) compromise. Purgy (talk) 07:54, 11 October 2015 (UTC)
 * One of the most frequent complaints that readers of these pages have is that they seem to be written by mathematicians for mathematicians. And when I look at some of the stuff we put out there, I can't really disagree with them. I don't believe in dumbing down articles as a cure for this, but I do think you have to be cognizant of the level of ability of your readership. As I've stated above, I don't think that the majority readership of this article is very mathematically mature, so I would try to keep the language plain at the expense of some precision and would sacrifice some conciseness to make things easier to follow. As to the specific edit I made that you objected to; I think the reaction of a typical reader who was not familiar with different representations of the natural numbers to the "In some representations" statement would more likely be confusion ("What are they talking about?") rather than your hoped for, "Oh, what a nifty idea. I must find out more about this." The real problem, as I see it, was that the sentence was being overloaded - more than one new idea was being thrown at the reader simultaneously. The solution would be to separate the ideas, that is, bring up new representations in a new sentence (and not in this paragraph). This would be a legitimate thing to do in the lead since there is material about this in the article and it would get an actual "pointer" as opposed to an inference of its existence. Bill Cherowitzo (talk) 19:16, 11 October 2015 (UTC)


 * I'd prefer a solution without sacrifice of precision, by just better linguistic means. But OK, let's agree to disagree about the level of ability of the majority of the readership and about the amount of precision and conciseness to be sacrifized for targeting the lower level within a sentence still below 40 words (including some marginal idiomatics). To make my point still more clear, I also object to blowing up this small remark about (mathematical) fallacious use of the notion of number beyond the current level. This is just my personal view of things, without any factual consequence. :) Purgy (talk) 07:40, 12 October 2015 (UTC)


 * ... and another suggestion, avoiding "representations":


 * Sometimes character strings that look like denoting natural numbers, are just convenient names (identifiers), called nominal numbers in linguistics, forgoing the properties of numbers in a mathematical sense. (~30 words) Purgy (talk) 09:29, 12 October 2015 (UTC)

Link needed
The term "Cauchy sequences" in the intro should be linkified. Thanks. 71.197.166.72 (talk) 19:19, 16 November 2015 (UTC)
 * Hmm &mdash; I'm on the fence between wikilinking it and just removing it (replacing it by something more vague). It seems more about the real numbers than about the natural numbers; probably too detailed for the lead of this article (though it could reasonably go in the body). --Trovatore (talk) 19:22, 16 November 2015 (UTC)


 * From my perspective of viewing at the integers as an important root of mathematical development wrt the manifold concepts of numbers, I'd like to keep these hints on means of extensions, be it by Cauchy sequences or by square roots or by inverse elements. Maybe the intricate details of the processes of extension are beyond the assumed average of math literacy of expected readers, and so linking to Cauchy sequences might overburden some, nevertheless, I plead for keeping these hints, be it as links, rather than removing them. Purgy (talk) 10:01, 17 November 2015 (UTC)
 * I think that would be reasonable in the body. I would frankly prefer not to have them in the lead.  Partly because of the difficulty of the material, but more because it strikes me as tangential to the subject of this article, which is the natural numbers, not the integers or the rationals or the reals.
 * For the lead, I would look for wording that mentions that these more complicated structures can be "built up" from the naturals, but without saying much about how. --Trovatore (talk) 17:19, 17 November 2015 (UTC)
 * It's fine if you want to move (or remove) the mention of Cauchy sequences, but wherever it appears first, it ought to be linkified. The worst possible combination is to leave the term in the lead and NOT link it.71.197.166.72 (talk) 05:55, 19 November 2015 (UTC)
 * Hmm, yeah, that's true. But there's no huge hurry about it.  We can decide which we want to do, and then link it wherever it appears, if it does. --Trovatore (talk) 07:39, 19 November 2015 (UTC)

I thought adding the links does not hurt anyone, fits to not only my preference and is no prejudice to the desires of Trovatore. -- Purgy (talk) 15:56, 19 November 2015 (UTC)

Problem with notations : the notations of this page do not respect international standard iso notation
The notation of this wikipedia page does not respect official iso notation.

As is defined in international standard ISO 31-11 :

The set of natural numbers; is the set of positive integers and zero

It is denoted by ℕ = {0, 1, 2, 3, ...}

Exclusion of zero is denoted by an asterisk: ℕ* = {1, 2, 3, ...}

This iso notation should be clearly said to be the international official notation in this page.

Other notation should be considered as non classical and non official notation sometimes used by some people.

This is important that wikipedia respects official international notation and does not propagate inofficial out of fashion notation.

--31.39.233.46 (talk) 19:45, 28 September 2016 (UTC)
 * ISO has no authority over mathematics. The fact that they even try is ... pathetic.  What we try to do is report the usages that actually exist in the mathematical community, including the situations where they are not always consistent among different workers.  --Trovatore (talk) 20:14, 28 September 2016 (UTC)


 * I do not want to generate the faintest impression that I would not appreciate to the highest level the standardization efforts of the ISO, but any effort to force mathematical truths, handled by the leading actors in different variants, into normed terms is ... yes, pathetic, and I consider this effort not noteworthy within an encyclopedia. I also do not agree to calling either use of the term natural numbers an inofficial out of fashion notation.


 * Furthermore, I am convinced that a pure mathematical view, focused on Natural Numbers, would neither refer to specific numerals, nor to the subsequent notion of integers, and especially not to specific objects like 0 and and 1, which get their important meaning as units wrt to two operations in algebraic structures built upon the naturals.


 * The cited standardization is not apt to unbiasedly reflect the basic defining properties of Natural Numbers, as widely employed within the mathematics community, but belongs to a specific, not unanimously shared convention, prevalent in only certain math topics.


 * Reflecting the status quo of use in Wikipedia is nobler a task than blindly adhering to this standard, imho. Purgy (talk) 07:33, 29 September 2016 (UTC)


 * What ISO 31-11 says (about “ mathematical signs and symbols for use in physical sciences and technology”) is part of the status quo and worth a small mention, which I have just added. It is perhaps unfortunate that our article on the superseding standard ISO_80000-2 does not include this specification, so we cannot refer to it. Of course individual mathematicians are not bound by the ISO! PJTraill (talk) 13:40, 29 September 2016 (UTC)

Repetition
On 26-3-2017 I did suppress the last parenthesis in “the integers, by including (if not yet in) the neutral element and an additive inverse (−n) for each natural number n (and zero, if it is not there already, as its own additive inverse)” as being a repetition, disturbing because one wonders if it really adds something… So, the whole of the parenthesis is undesirable. Surprisingly, somebody undid my edit 15 minutes later. Would somebody please explain why I am wrong… or shall we suppress the parenthesis “(and zero, if it is not there already, as its own additive inverse)”? --Dominique Meeùs (talk) 12:48, 28 March 2017 (UTC)
 * because after “including (if…) the neutral element” zero, there is no question whether it is there or not, it is: is has been included; thus a second “if” as “zero, if it is not there already” does not make sense any more;
 * because “its own additive inverse” is evidently implied by “neutral”.
 * I agree, and I have edited again the article: Only the added numbers must be described, not their properties (here the fact that –0 = 0). D.Lazard (talk) 18:24, 28 March 2017 (UTC)
 * According to the history listing I was the one reverting and apologise for it. I must have been sleeping. I somehow was under the false impression that you deleted the inclusion of zero (as happened some time earlier).&minus;Woodstone (talk) 16:19, 30 March 2017 (UTC)

Rationals and reals
The natural numbers are the basis from which many other number sets may be built by extension: the integers, by including (if not yet in) the neutral element 0 and an additive inverse (−n) for each nonzero natural number n; the rational numbers, by including a multiplicative inverse (1/n) for each nonzero integer n; the real numbers by including with the rationals the limits of (converging) Cauchy sequences of rationals; 

That last part is wrong: the multiplicative inverses of nonzero integers form a very limited subset of the rationals (roughly: it omits any fraction with a numerator that isn't 1), and the only limit of (nontrivial) Cauchy sequences in that set is 0. It's been a long time since I had to know all this stuff, but my dim recollection is that the rationals are surprisingly difficult to define succinctly—something about equivalence classes of pairs of integers—so I'm not sure what I would suggest as replacement text, but as it stands it's obviously incorrect. --2001:1970:4F68:E000:C876:D11D:CB37:83F6 (talk) 10:39, 25 May 2017 (UTC)
 * MOS:MATH says "The lead section should include, where appropriate ... an informal introduction to the topic, without rigor, suitable for a general audience. Here, the lack of rigour consists of omitting that rational numbers include other numbers. Nevertheless I'll try to fix this. D.Lazard (talk) 10:59, 25 May 2017 (UTC)

Natural numbers used for ordering?
Are natural numbers really used for ordering? Is third a natural number? Numerals representing specific natural numbers can be used for ordering, but can natural numbers? 83.230.4.140 (talk) 15:54, 4 October 2017 (UTC)


 * I strongly object to your claim that the properties of numerals, without referring to any ordering principle from outside of this notion allow for using them for ordering purposes. It is the defining structure of successor that makes the naturals suited for ordering. Maybe, one can debate if this is the unique origin of "ordering", but it is a quite general one and apt for broad application. I claim that numerals, whenever employed for ordering purposes, inherit their ordering power from the successor principle of the natural numbers, and cannot establish any order on their own. The cited ordinals (third, fourth, ...) are obviously derived from their linguistic counterparts (three, four, ...), denoting natural numbers, and "second" ("sequitur") directly hints to the "successor" of the "first". Purgy (talk) 09:12, 5 October 2017 (UTC)

I don't see any claim in the above post, only three questions. I think your answer, Purgy, is at a higher level than the question asked. To answer the users questions. 1) Yes, the natural numbers are used for ordering. The naturals numbers are one, two, three, and so on.  "Third", as in "first", "second", "third", are adjectives, which describe the position of something in an ordered set, e.g. "first person in line", "second person in line", "third person in line".  A numeral is just a symbol for a number, and we use symbols to communicate ideas about numbers.  I hope this helps. Rick Norwood (talk) 19:04, 11 December 2017 (UTC)


 * Rereading my reply I perceive a not fully intended harshness &mdash;apologies. I certainly then felt a "claim" about "numerals are for ordering, naturals are not", induced by the "really" in the first question and the first part of the last sentence, undeniably being a claim. This leads me to the consequence that the level, as well as the attempted strictness in my reply may not be fully irrational. E.g., are naturals themselves "really" useful for ordering, or isn't it their axiomatic property of having exactly one successor, that is transferred to establish the ubiquitous "numerical" ordering? Numerals require still one more step of transfer, imho. In any case, the replies hopefully suffice to answer the "biased questions" on various levels. :) Purgy (talk) 07:59, 12 December 2017 (UTC)


 * "Really" must be used with care, because of it strong epistemologic implication. Here, the question and the answers suppose implicitly that numerals and natural numbers belong to the "reality". In fact, this is not the case, numeral first, and natural numbers much later have been invented for expressing properties of the "real world", which are cardinality and succession (one after the other). Natural numbers have been invented for allowing working with numerals. For example, in a date, the year is an ordinal numeral (2017th year after Christ), but for computing your age, you need a subtraction, which consists of manipulating ordinal numerals as natural numbers.
 * Thus my answer to the original question is: (ordinal) numerals, such as "third" are symbols that have been introduced for ordering (and counting, in the case of cardinal numerals). Thus numerals are not natural numbers. Natural numbers have been introduced for modeling (and later for formalizing) the experimental properties that result of ordering and counting. One of these properties is that a unique natural number may be associated to each numeral, and thus that in turn, natural numbers may be used for ordering and counting.
 * Thus, behind this very elementary question are hidden the very deep questions of what is mathematics, and what is its relationship with reality. My preceding answer reflects my personal views on this question, and these views are not shared by all mathematicians. D.Lazard (talk) 09:08, 12 December 2017 (UTC)
 * Just to confirm that not all mathematicians agree on these things, I think the natural numbers were discovered not invented ;-) Paul August &#9742; 13:02, 12 December 2017 (UTC)

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

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

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

Whole numbers = integers?
The article says "in other writings, the term (whole numbers) is used instead for the integers (including negative integers)", but it gives only one citation, and even that citation is defining "integer" to mean "whole number", not vice versa. I suspect that including negative integers in the whole numbers is nonstandard, a convention used only by a fringe. Does anyone have references to suggest otherwise? Ebony Jackson (talk) 04:03, 25 January 2021 (UTC)
 * At the very least "that term" is unnecessarily ambiguous. I suspect it refers to "whole numbers". The whole sentence could use some cleanup. There should be a separate reference for each of the uses. BFG (talk) 14:58, 25 January 2021 (UTC)
 * (Heavy sigh) This again. I really wish we could just ban the term "whole number" and not let it appear anywhere in Wikipedia.  That would fix the issue good!  And nothing would be lost in terms of modern serious mathematics, because the term "whole number" is essentially unused in that context.
 * Unfortunately it is used in other contexts, and with inconsistent meanings. The one I learned in California K-12 education was that the natural numbers exclude zero and the whole numbers include it.  Since the more useful definition of "natural number" includes zero, that one is superseded.
 * However it seems that other writers use it to mean natural numbers excluding 0, or integers (including negative). See the archives of talk:Whole number to find tedious discussion on this point, possibly with refs. --Trovatore (talk) 20:43, 25 January 2021 (UTC)
 * Update: Just looked through that talk page and its one archive and nothing jumped out at me as useful.  Maybe the archives of this talk page, or of talk:integer?  I know it was somewhere. --Trovatore (talk) 20:45, 25 January 2021 (UTC)
 * OK, this may be what I was remembering: talk:natural number/Archive 2. It seems that  particularly was arguing for the existence of the "whole number==integer" meaning; if he/she is still around, perhaps we will hear more.
 * I see that Maproom presented two references for this usage, neither of which is really ideal; one was a dictionary, and the other was Mathworld. Honestly I would be inclined to discount Mathworld almost completely on issues of usage.  Mathworld is a reasonable site for mathematics, but really an appallingly bad one for nomenclature.  And both Mathworld and dictionaries are tertiary sources, which are not as good as secondary sources.  That said, it seems to me that there's very little harm in saying that "whole number" can sometimes have the sense of "integer", given that (1) it seems to be true, for some value of "sometimes", and (2) the term "whole number" isn't used much in the serious mathematical corpus, so it doesn't matter too much if people allow it meanings that it rarely has. --Trovatore (talk) 06:18, 26 January 2021 (UTC)
 * If I once had an opinion on this, I've forgotten it. I would recommend writing "integer", "non-negative integer" or "positive integer", and avoiding "natural number" and "whole number" as ambiguous. If the article is to mention the ambiguous nature of terms, it should supply references – they can't be hard to find. Maproom (talk) 08:17, 26 January 2021 (UTC)
 * Just to give some perspective on this: The German word for 'integer' is 'Ganze Zahl', which translates to 'whole number', so I could see how this could cause some confusion, at least for a non-native English speaker like me. :) Phonous (talk) 23:41, 4 April 2021 (UTC)
 * In French also, the word for 'integer' is nombre entier often abbreviated as entier, whose literal translation are 'whole number' and 'whole'. The term for 'natural number' is nombre naturel or entier naturel; the first term translates literally to the English, and the second one could be translated 'natural whole' or 'natural integer'. The term entier relatif ('relative integer') is also used for "integer", for emphasizing that negative integers are included. All of this suggests that the term "whole number" results from the influence of other European languages, and this may explain its different uses. D.Lazard (talk) 08:03, 5 April 2021 (UTC)
 * The use of the phrase "whole number" is just one more example of the disconnect between K-12 usage and professional usage in the United States. I do not know if this is reflected in other English speaking countries. I have never heard a STEM professional use the phrase "whole number". Professionals either use natural number or integer. But K-12 teachers, taught the phrase "whole number" themselves, insist on teaching it to their students. If Wikipedia mentions it at all, it should say, "Whole number" is a phrase used in K-12 education with inconsistent meanings, some of which include the meaning "Natural number", "non-negative integer", and "integer". Rick Norwood (talk) 11:17, 5 April 2021 (UTC)
 * Just weighing in I corrected the page Whole number to mention that it is a colloquial term. points out the word in German is 'Ganze Zahl' which roughly translates to whole number. The use in German should carry some weight, as both $$\mathbb{N}$$(Natürliche Zahl) and $$(\mathbb{Z})$$(Ganze Zahl) takes their letter from German. This nomenclature is reflected in all the Scandinavian languages, but in French the names translates to literally natural integers and relative integers. I would claim that the use of whole number thus should mean the integers $$(\mathbb{Z})$$. But that is just my personal opinion, I think it's better to only talk of it as a colloquial term. BFG (talk) 12:46, 5 April 2021 (UTC)
 * I did a google ngram search comparing the frequency of "natural numbers" versus "whole numbers" and it appears that up to about 1960 "whole" was rather more frequent, and "natural" started somewhat dominating since only about 2010. So whatever meaning is attached to the terms, above conclusions do not seem warranted.&minus;Woodstone (talk) 13:28, 5 April 2021 (UTC)
 * I don't think that's a valid comparison. This is a mathematics article and should give priority to the usage in serious mathematics.  A start might be to limit the search to Google Scholar; not sure if that's feasible with ngrams. --Trovatore (talk) 17:09, 5 April 2021 (UTC)

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


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

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

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