Talk:Natural number/Archive 1

Untitled
This is the archive "Talk:Natural number/Archive 1". It covers talk up to the end of 2006.

Math article protocol
''Wikipedia follows this convention, as do set theorists, logicians, and computer scientists. Other mathematicians, primarily number theorists, often prefer to follow the older tradition and exclude zero from the natural numbers.''

I have used similar phrasing in other articles, and had it changed, saying "Wikipedia does not refer to itself" or "don't refer to Wikipedia." In mathematics, this would seem to rule out any attempt at universal article terminology protocol. Is this the intended consequence? I'm curious to know, for math articles. Revolver 23:50, 9 Jun 2004 (UTC)

As Wikipedia is edited by (loosely) the world, it's a little inaccurate to say "the convention used in Wikipedia" in the same way you would say "the convention used in this book", when you neither have checked all the articles in Wikipedia or approve all edits made (or watch all math pages so you can "correct" them). I also see no reason to follow convention arbitrarily set by someone who pretends (s)he can predict the future of Wikipedia (namely, that Wikipedia will include 0 in the natural numbers). Perhaps we should call a vote. --Elektron 14:52, 2004 Jun 12 (UTC)

Peano axioms
I made some substantial changes to the Peano axiom stuff. I hope this makes things more clear, and hopefully, it should also clear up the "debate" about whether 0 is a natural number or not. In essence, we're either talking about the "natural numbers" as a label or term to identify either the set {0, 1, 2,...} or {1, 2, 3, ...} whether you have your set theory or number theory hat on, OR you mean it in terms of Peano axioms. The former sense is already discussed (i.e. the "debate" between what is meant by different people is mentioned) in the first part of the article, so I think the rest should focus on the second. Otherwise, there would be no distinction between "natural numbers" and "countably infinite set"...it's the addition, multiplication, order that we care about!

So, this completely changes the "debate" about the status of "zero". The debate that has raged on this talk page largely centers around use of the term to indiciate a set of integers, usage, in other words. That's the less important sense of the term. In the more important sense (Peano axiom sense) the term "zero" simply means any object, that when combined with a successor function, satisfies the axioms. It's like talking about the "zero" of a group. This is why it is perfectly reasonable to take

N = {5, 6, 7, ...}

"0" = 5

successor of a = a + 1

as a system that satisfies the Peano axioms, even though 5 isn't equal to "0" in the normal sense of the number.

Of course, once this possibility is mentioned, we can agree to take the usual construction of the natural numbers set-theoretically from that point on. But not doing so introduces problems in logical dependence:


 * It doesn't make sense to talk about "a + 1" as was talked about in the Peano axioms, when the operation of addition hasn't even been defined yet. Nor has the number "1" even been defined yet! You're really talking about the successor function, and skipping ahead of yourself mentally. This confuses things, because it implicitly assumes the existence of N that satisfies the axioms, which is just what you're supposed to be proving.
 * You might take the usual construction of N and define the order by saying one number is ≤ another if it is a subset of it. Again, this skips ahead of yourself -- the order should only be defined in terms of 0, the successor function, and anything defined in terms of them. This is done correctly here, but it's still a bit confusing earlier when things are said like "the set n has n elements"...this is actually a triviality, if you define cardinality in the normal sense -- having n elements literally means being able to be put in 1-1 corr. with the SET n, also the thing about ordering defined by subsets -- this could also be true by definition, since ordering of ordinals is often defined in terms of the subset relation.

Revolver 02:58, 13 Mar 2004 (UTC)

Zero
While I agree with the sentence that many authors have historically excluded 0, would it be possible to add a sentence saying that in this encyclopedia, we always include 0? That way, we can unambiguously use links to natural number whenever we mean "non-negative integer", an awkward term. --AxelBoldt

I disagree. A similar difference of opinion occured here at wikipedia with regard to "ln" versus "log" and it was determined that "ln" should be used because it was unambiguous (although, to my dismay, it was apparently agreed that whenever base 10 was used for "log", it need not be explicitly mentioned or symbolised...as a number theorists, I interpret "log" to mean "natural log" by default, and I have to be reminded if it's otherwise.)

So, why not make it a policy to always use the unambiguous terms


 * positive integer
 * nonnegative integer (why do we hyphenate, anyway??)
 * negative integer
 * nonpositive integer
 * integer

instead of "natural number" or "whole number"? I read a lot of stuff in BOTH set theory AND number theory, and it is the convention in the former to include 0 and in the latter to exclude 0 from the definition of natural number. There are good reasons to justify these decisions in each field -- one definition is not more "correct" than the other, it's just that in set theory, 0 makes conceptual sense to include, while in number theory, it makes similar conceptual sense to exclude it. Thus, ANY choice of definition for natural number is going to go against SOMEONE'S convention. So, why not choose to use the unambiguous terms? I don't really believe they're that "awkward", esp. compared to many other math terms, and more importantly, they're precise and eliminate confusion.

BTW, the term "whole number" is rarely used by any mathematicians, in my experience, and I've never seen the symbol "W" used for it, whatever it's intended meaning.

Revolver 14 Jan 2004

Just as a note, I'm an undergrad math student at UC Berkeley and I know I'd get marked incorrectly for including 0 in N. For us, N is {1,2,...} and the set {0,1,2,...} is W. Whether or not this is good or bad is obviously an issue of contention here but I wanted to add what has been institutionalized here. I came here to brush up on mathematical induction but am having some difficulty because I have been taught that induction starts with the base case n=1...I guess it shouldn't be a big difference, or should it? Goodralph 04:46, 20 Feb 2004 (UTC)


 * If you'd get marked wrong for including 0 in N, that probably just means you're taking a number theory or algebra class, not a set theory class. I seriously doubt that Berkeley as a department has some official "policy" that's been institutionalised on the definition on N; if you looked up papers by all faculty members, you'd find papers that use one definition and other the other.  As for where to start induction, you can start at any integer you want, 0, 1, -1, -57, 83, etc., since every {k, k + 1, k + 2,...} is the same as a well-ordered set. Revolver 00:24, 21 Feb 2004 (UTC)

First, "0" is not 'natural' to start with, since you can't get something from nothing (you can't get 1 from 0). Naturally, we define u0 = 1, and un+1 = un + u0. We only need to have three things already defined: =, 1, +; whereas starting at 0 requires =,0,1,+.

I also vote to abandon use of 'natural number' (I've always used it to mean "positive integer", and it's only meant that in all the books I've seen). Elektron 05:15, 2004 May 8 (UTC)


 * While in theory it would be nice to have one unambiguous notation or terminology for every concept in all of mathematics, in practice it just doesn't work that way. Mathematicians, physicists, engineers, and others in different fields like to used slightly different notations for concepts that are actually the same.  It is pointless and counterproductive to insist that all these diverse groups use consistent notation.  It is like insisting that everyone in the world speak the same artifically constructed bastardized dialect.  Why can't Wikipedians just use the notation and terminology that is standard in the field they are writing about and not try to force all articles with mathematical content to conform to some completely artificial universal notation? If you feel notation is not clear from context, just explain briefly what is meant by it in that particular article.  Personally I don't care how you use symbols such as, to pull an example out of the blue, $$\subset, \subseteq, \subsetneq$$, as long as you take the time to explain briefly what you mean by them if it is not clear from context.  Just don't fall into that irritating habit of using the most contrived notation and terminology possible "because it's unambiguous," and then "correcting" people who use slightly different notation than you do, and who may actually be more correct than you are.  -- 130.94.162.64 20:41, 30 November 2005 (UTC)


 * Just to make it clear: when you are talking about Peano axioms and set theory, 0 is a natural number. When you are talking about number theory, it is not.  Don't try to force one field or the other to use unnatural terminology or notation.  This reeks of Orwell's newspeak. -- 130.94.162.64 20:47, 30 November 2005 (UTC)

The article currently states that + is sometimes added as a subscript or superscript to Z to denote positive integers, but that it is also commonly used for non-negative integers. However, is this really true for the subscript case? The example given is superscript, and I've only seen superscript used in this way. Is $$\mathbb{Z}_+=\{1, 2, 3, \ldots\}$$ actually unambiguous? Coffee2theorems 06:25, 21 August 2006 (UTC)


 * Unhappily, no, as witnessed by examples referred to in this section and the following. However, it is almost impossible to formulate a mathematical terminology, such that you cannot find anyone employing the terms differently.  I have once seen an author, in whose opinion the positive integers were 0,1,2,3,..., and the nonnegative integers were 1,2,3,...; and if my memory is correct, (s)he also used $$\mathbb{Z}_+$$ for 'the set of the positive integers', in his/her sense.  (There is nothing illogical with this usage; it just presupposes that you consider zero as both positive and negative, instead of neither.  Irritating and exasperating, yes; but not illogical.)


 * I just noted that part of this 'irritating and exasperating' terminology actually may be blamed on 'my heroes', the Bourbakists; see Elements of mathematics; Algebra; I,§2,5, where they actually do define 0 as both positive and negative. In the second French edition, there is a note, where they explain this deviation from standard notation by a wish to adapt to the usage in the theories of sets and of ordered groups. JoergenB 14:47, 16 September 2006 (UTC)


 * The article seems to have reached a reasonable balance, as it now appears. I am going to add the fact that the Bourbakists decided to include 0, since it is relevant for many other mathematicians than me.  We are a bunch who prefer their choices of terminology, all else being equal, since they really made an effort to fix a common terminology (valid for all branches of mathematics).  Thus, N includes zero, also in e.g. Bourbaki texts on algebra. JoergenB 13:56, 16 September 2006 (UTC)

Is "positive integer" also ambiguous?

 * I think "positive" is just as ambiguous: does it include zero or not? -- Jitse Niesen 14:07, 16 Jan 2004 (UTC)
 * "Positive" definitely does NOT include zero. There's no doubt or ambiguity about that, either in English or mathematics. I cannot speak for other languages. Peak 07:21, 17 Jan 2004 (UTC)


 * http://thesaurus.maths.org (mathematical thesaurus, maintained by the University of Cambridge) says "It is not universally agreed whether this set contains zero or not. It is better to use the terms strictly positive and non-negative to indicate whether zero is to be included or not." (see http://thesaurus.maths.org/dictionary/map/word/1011). Indeed, Google finds 55600 pages with "strictly positive", a phrase which would not make sense if there were no doubt whether "positive" includes zero. --Jitse Niesen 13:18, 20 Jan 2004 (UTC)

[Peak:] Firstly, the page you refer to is about the REAL numbers, not the INTEGERS.

Secondly, the quotation you give is incomplete. The first paragraph states unambigously: "The set of positive real numbers ... contains all real numbers greater than zero." (There is no implication here that it might also contain 0, any more than there is the implication that it might contain the negative numbers.)

Thirdly, the people in Cambridge are evidently being very polite. Instead of saying, "Some people are confused...", they said "It is not universally agreed..." If some people really insist that the word "positive" means precisely ">=0" I don't really see why they would accept "strictly positive" to mean anything other than "strictly >= 0", using the ordinary meaning of "strictly" (i.e. "without exception").

Fourthly, for Wikipedia, to determine the commonly accepted meaning of a word, it is best to go to good dictionaries such as the American Heritage Dictionary, the online version of which states:


 * 11. Mathematics a. Relating to or designating a quantity greater than zero.

Finally, I suspect that you'll find that the frequency of the phrase "strictly positive" has nothing to do with confusion or ambiguity about the meaning of the phrase "positive natural numbers." The first page of Google results that I got had references to real numbers, datatypes, and operators. Even here, the use of the word "strictly" often is for emphasis (i.e. meaning "no exceptions"), as in "He's a strict vegetarian" (i.e. he adheres to the rules strictly). Peak 06:39, 21 Jan 2004 (UTC)


 * I think "strictly" in "strictly positive" refers to the strict inequality x > 0 as opposed to the inequality x >= 0. See for instance Bernoulli inequality for this meaning of strict.


 * However, I do agree that most mathematicians, and most people in general, use "positive" to mean "> 0", and I think Wikipedia should also use it in this sense. The only thing I do not agree with is that it would be clear to everybody that "positive" means "> 0". When I read "positive", I think: this probably means "> 0", but there is a small chance that the author actually meant ">= 0". On the other hand, when I read "natural number", I think: this can either include or exclude zero, and I have to check which definition the author uses if it matters. -- Jitse Niesen 12:15, 21 Jan 2004 (UTC)


 * Some points to make:
 * I don't know anyone personally who considers 0 to be a positive number. So, just from my own experience in math, considering 0 not to be positive is an almost universal "convention" (although it's not just convention, see next point).
 * The definition of "set of positive elements" of an ordered ring in ring theory specifically excludes 0 as an axiom, for good reasons (essentially, we want to have trichotomy). So, excluding 0 for the integers conforms to this definition.
 * I think the disclaimer above about "no universal agreement" really is a euphemism for "some people are mistaken", or possibly a cavaeat emptor, because of point 1.
 * The use of the word "strictly" is used for inequalities to mean "and not equal", e.g. in a poset, the order relation is usually taken to be "less than or equal to", so you have to explicitly mention that equality is being ruled out. This makes a LOT of difference, esp. in areas like analysis, where the difference between possible equality and strict inequality is crucial. But "positive" already has the strict inequality built into it, since 0 is excluded in the definition. Saying "strictly positive" is not wrong, it kind of emphasises it, but it's not necessary.
 * Most of the google hits had nothing to do with the integers or the real numbers. In other areas of math, they might have their own definition of "strictly positive" (it seems to pop up in Hilbert spaces and operator theory a lot), but that has nothing to do with the integers.
 * Revolver


 * There's no ambiguity when someone says "12 and under" or "under 12". "positive integer" is also the best way to say "n &ge; 1, n &isa; Z". "strictly positive" means nothing when the reader doesn't know what positive means, just like "integers greater than 0" doesn't mean much when the reader doesn't know the meaning of 'greater than'. I mean, we could say "integers greater than or equal to 1", but who really wants to do that? Elektron 05:05, 2004 May 8 (UTC)

Axioms
"Axioms should be minimal" is a fine statement which doesn't reflect the way mathematicians actually work. For instance, the commonly accepted list of axioms for a vector space is not minimal; nor is the set of axioms for a group. Nevertheless, nearly all sources define groups with the non-minimal and symmetric set of axioms rather the minimal and obscure one.

The same is true about the Peano axioms. It's possible to present a minimal version of them; yet these are not Peano's axioms as they're normally presented in mathematical texts. One very good reason to retain the axiom "every natural number except 0 has a predecessor" (which is in fact the commonly accepted form of this axiom) is that it's common to treat the induction axiom as a special and very strong axiom, and to study fragments of Peano arithmetic defined by other axioms without induction, or with weaker forms of it. -- AV

I added a note after somebody had modified (vandalized) the first axiom to say "there is not a natural number 0". And there are some that consider 1 the first natural, so I added a note.--AN

This system of axioms is too weak, isn't it? To be specific, it allows the construction of what Douglas Hofstadter calls "ω-inconsistency" -- that is, you can define a property X and state that some natural number possesses X, when in fact there is no such natural number. And you still have a consistent system. --User:Juuitchan


 * Can you (1) prove from the axioms that some natural number possesses X, or can you (2) just not disprove it? I would be very surprised if (1) were the case; on the other hand, (2) is unavoidable and is achieved by several incompleteness results. AxelBoldt 04:42 Nov 19, 2002 (UTC)


 * I am referring to your (2). What I want is an axiom that says this: For any natural number a, if you count up like this: 0, 1, 2, 3, etc., by ones, mechanically, like an odometer, you will sooner or later REACH a. I have a feeling that this axiom, while easy to understand, is impossible to formalize, and this is because the axiom depends on the notion of time, which is completely foreign to mathematics. NB: It might help to understand that I, like many others, rely on pictures as mental models for abstract concepts. My mental model for the natural numbers is as follows: Think of an odometer wheel showing zero. Now this wheel can count forwards and backwards (just not backwards past zero. If it tries that, it will disappear in a puff of smoke.) This is a magical odometer wheel: push it forwards past 9, and it will "grow" another wheel and show 10, and you can keep going, 11, 12, and so on. Push it back past 10, and the extra wheel will disappear. Now, all these positions and forms that the wheels can take correspond to the natural numbers. This is MY model, not some crazy formalization that allows for supernatural numbers. With my model, the supernaturals are impossible: no matter how far you push the wheel, you will never get a supernatural. --User:Juuitchan


 * Your axiom is equivalent to Peano's last axiom (the axiom of induction). Think about it this way.  Obviously 0 is reachable by counting up from 0.  Now, if a natural n can be reached by "counting up" from 0, then n + 1 can also be reached by "counting up" from 0 -- we just did it. And, therefore, by the axiom of induction, all natural numbers can be reached by "counting up" from 0.


 * No, it isn't. You're assuming that the intuitive property 'can be reached by counting up from 0' can be expressed as a formal proposition, P(n), say, so that the set {n is an element of N: P(n)} can be formed.  By the indictinve axiom, this set is all of N.  However there is no way to formaulate this intuitive property, other than by the Peano axioms.  I.e., we assume that it applies to all n in N. -- Daran 11:41, 7 Oct 2003 (UTC)


 * I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0. So Juuitchan's intuition turns out to be like the parallel postulate in Euclidean geometry.  However it appears to be the case that, if you define a set W recursively as { 0 in W; a in W -> S(a) in W } where S(a) is the successor function (+1), then N contains W.  In other words, all numbers that can be reached by counting up from 0 are natural numbers, but not necessarily the reverse.


 * Wikipedia doesn't have a nonstandard arithmetic article. I would write one but I don't know enough about the subject and casual web searches don't turn up enough material.  -- Zack 19:52, 10 Oct 2003 (UTC)


 * I don't know what you mean by "supernaturals". Infinite ordinals, perhaps?  But no one tries to argue that the infinite ordinals are natural numbers. -- Zack


 * You'll never get the entire set either. You model allows for any finite number, however large, but not for the set of all of them.

Whole number
I put back the text discussing the meaning of the term whole number. This term has a disputed meaning -- I don't question that some people do use it to refer to the integers, but others do use it in the way I originally described. The set letter W is invariably used as I described. I've tried to explain it a bit better this time. An alternative would be to drop this text entirely and discuss the disputed meaning under Whole number.


 * Zack 21:54, 3 Oct 2003 (UTC)

I gave the three meanings for "whole number" here for now, without picking sides. I have never seen W for the set of whole numbers, but I'll leave it in. I also removed a non-standard construction of N which will only confuse the reader. AxelBoldt 21:26, 6 Oct 2003 (UTC)

It looks good. I'm going to do some copyediting in a bit.
 * Zack

Miscellany
Wouldn't the follow excerpt from this article be more correct of the positive integers or counting numbers? I agree that zero should be included in the set of natural numbers, but zero is not among the first numbers learned by children, and arguably conceptually more difficult to learn than the counting numbers.


 * "These are the first numbers learned by children, and the easiest to understand. Natural numbers have two main purposes: they can be used for counting ('there are 3 apples on the table'), or they can be used for ordering ('this is the 3rd largest city in the state')."

--Sewing 18:28, 18 Dec 2003 (UTC)

All said and done, the unfortunate truth is that some sources call zero a natural number and others don't. Tmesipt. 2.20.04.

Better axioms
It's a little stupid to define "natural number" = "positive integer" when you have no definition of positive integer (unless we define "natural number" in terms of the integers). But we can define natural numbers thus:
 * 1) There is a natural number 1.
 * 2) For every natural number n, there is a natural-number successor S(n) > n.
 * 3) No other natural numbers exist.
 * 4) n + 1 = S(n)
 * 5) a + S(b) = S(a + b).

--Elektron 15:08, 2004 Jun 12 (UTC)
 * We can't define succession as the addition of unity before we define addition, so "a + S(b) = S(a + b) for all a, b" doesn't define addition when we define the successor in terms of addition.
 * "No other natural numbers exist" iff "If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers." and is much simpler anyhow. I think it should also imply the predecessor axiom (unless we need another one that says if a > b and b > c then a > c).
 * Allowing natural numbers we can't count up to is useless unless ∞ is a natural number, and ∞ + 1 ≠ ∞.


 * 1) For every natural number n, there is a natural-number successor S(n) > n.
 * What does ">" mean?


 * 1) No other natural numbers exist.
 * Exist besides what? This statement doesn't have meaning for me. It either seems nonsensical or tautological, in neither case is it an axiom.


 * 1) n + 1 = S(n)
 * Is this a definition or an axiom or what? I don't follow.


 * 1) a + S(b) = S(a + b).
 * Again, is this defining +? You don't have to define + to give axioms for N.


 * We can't define succession as the addition of unity before we define addition, so "a + S(b) = S(a + b) for all a, b" doesn't define addition when we define the successor in terms of addition.
 * Succession isn't defined in terms of addition. It isn't defined in terms of anything. It's just some function satisfying the axioms. Addition has nothing to do with it.


 * "No other natural numbers exist" iff "If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers." and is much simpler anyhow.
 * The former statement is nonsensical, though. Or too vague. (See above.) The latter is precise.


 * I think it should also imply the predecessor axiom (unless we need another one that says if a > b and b > c then a > c).
 * Again, what is ">"?


 * Allowing natural numbers we can't count up to is useless unless ∞ is a natural number, and ∞ + 1 ≠ ∞.
 * I don't follow. Are you suggesting to change the axioms? They are quite standard and correct as stated. Revolver 10:18, 13 Jun 2004 (UTC)

I'm suggesting that the Peano axioms aren't really axioms (in the article, they're called postulates). If you define the set of natural numbers N = {1,2,3,4,...} ∪ {±1/2, ±3/2,±5/2, ...}, they satisfy all of the postulates except the mathematical-induction postulate (If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers), and proof that they don't satisfy this isn't easy when you, as above, don't allow the axioms to define &gt; or &lt;. "No other natural numbers exist" is an icky way to disallow these. If you do allow use of &lt; in the axioms, then you can say "there is no natural number a which satisfies n &lt; a &lt; S(n) for all natural numbers n", but then you need to disallow {∞ ± n: n &isa; N} from the set of natural numbers (in the Axioms section, I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0.), which would not satisfy the mathematical-induction postulate either.

I also see nothing wrong with disallowing (implicit or otherwise) definition of, say, {+,&gt;,=} in the axioms. --Elektron 07:13, 2004 Jun 18 (UTC)

Elektron, I'm not an expert in model theory or anything; my working experience is within the cozy confines of ZFC (really, one can get a ph.d. in math without being exposed to any set theory or logic at all), but I think you're bringing in extraneous issues. Whatever philosophical or model-theoretic issues surround the axioms is worth putting in the article, but this hardly changes the axioms themselves. Whether or not the second-order Peano axioms "capture" what we mean by "the natural numbers" is maybe something for logicians and philosophers to sort through; how does that affect the statement of the (second-order) axioms themselves?

I'm suggesting that the Peano axioms aren't really axioms (in the article, they're called postulates).
 * I think axiom/postulate are interchangeable terms. I don't know what you mean, "they aren't really axioms". They most certainly are, no different than the axioms defining a group, a topological space, or a geometry. The set of natural numbers with successor function satisfies the axioms, so they're consistent.

If you define the set of natural numbers N = {1,2,3,4,...} ∪ {±1/2, ±3/2,±5/2, ...}, they satisfy all of the postulates except the mathematical-induction postulate (If a property is possessed by 0 and also by the successor of every natural number which possesses it, then it is possessed by all natural numbers),
 * I'll take you to include "0" in N. And I assume your successor function is defined as "adding one" on {0, 1, 2, ...}, you don't mention how it's defined on the fractions. I don't disagree with what you say here.

and proof that they don't satisfy this isn't easy when you, as above, don't allow the axioms to define &gt; or &lt;.
 * Sure, it's easy. A = {0, 1, 2,...} contains 0 and is closed under the your so-called successor function, (no matter how you choose to define it injectively from FRAC = {±1/2, ±3/2,±5/2, ...} to FRAC), yet A is not equal to N = A &cup; FRAC. End of proof. I'm confused, you still seem to be assuming that the "successor function" is synonmous with "add one". It isn't..."successor function" just means "anything that satisfies the axioms".

"No other natural numbers exist" is an icky way to disallow these.
 * As I said before, the statement "no other natural numbers exist" is nonsensical -- could you please express it more precisely? No other natural numbers exist BESIDES WHAT? I really can't make heads or tails of this statement.

If you do allow use of &lt; in the axioms,
 * But then, they wouldn't be THE PEANO AXIOMS...they'd be something else. If you want to talk about other sets of axioms besides second-order peano axioms, fine. But it's not a matter of "allowing" &lt;, &gt;, or +, it's just that these aren't what we mean by "second-order Peano axioms", it's like saying "if we do allow * and / (mult and div)" in the axioms for a group...but the group axioms aren't about some hypothetical functions, only about the group operation. Similarly, the second-order peano axioms aren't about ordering or arithmetic operations, only about the SUCCESSOR FUNCTION.

then you can say "there is no natural number a which satisfies n &lt; a &lt; S(n) for all natural numbers n", but then you need to disallow {∞ ± n: n &isa; N} from the set of natural numbers
 * Again, you're ahead of yourself. When discussing the peano axioms, there is no such thing as "the set of natural numbers", this is just a particular example satisfying the axioms. You should be able to state the peano axioms without using the words "natural number". Similarly, I don't know why you feel the need to exclude "infinity", when "infinity" isn't mentioned in the axioms.

(in the Axioms section, I asked a mathematician friend and was told about "nonstandard arithmetic" which admits the existence of numbers n in N that are not reachable by counting up from 0.''), which would not satisfy the mathematical-induction postulate either.
 * "Counting up" is a pretty vague term. In fact, the whole point of the axioms via the successor function is to clarify what "counting up" means. Yes, there are such things as "nonstandard natural numbers" in nonstandard analysis (a la Robinson), but these are not considered to be elements of N, in fact, the definition of a nonstandard natural number is a member of *N not in N. If you're talking about nonstandard models of arithmetic, I talked about this above. There are certainly first-order models of the second-order axioms, such that every first-order statement in the second-order model is provable in the first-order model; this doesn't change the definition of the second-order axioms.

Revolver 11:39, 18 Jun 2004 (UTC)

''Sure, it's easy. A = {0, 1, 2,...} contains 0 and is closed under the your so-called successor function, (no matter how you choose to define it injectively from FRAC = {±1/2, ±3/2,±5/2, ...} to FRAC), yet A is not equal to N = A ∪ FRAC. End of proof.''
 * That at best proves that A is a set of natural numbers, and says nothing about whether 1/2 is one. Elektron 17:59, 2004 Nov 1 (UTC)
 * Again, you're confusing the axiomatic approach with an "essence" approach. There is no such thing as a "natural number". A "natural number" is precisely an element of a structure satisfying the Peano axioms. The question I was answering was not "is 1/2 a natural number?" It was, does the set N = A U FRAC satisfy the Peano axioms, and I showed that is doesn't. (Note that "N" here, isn't N = {0, 1, 2, ...}) We showed that this set N with this particular "successor" function violates the last axiom, because there is a subset A such that A contains 0 and A is closed under the successor function, yet A is a proper subset of N. This is not allowed. Asking the question "is 1/2 a natural number" is nonsensical and jibberish. Revolver 03:25, 2 Nov 2004 (UTC)

Hope this clarifies
It seems a lot of the confusion rests on how to interpret the "induction" axiom. In the natural number article, there is an "informal" list of axioms given. This list is just that -- informal, i.e. not precise. There are 2 different ways of interpreting them, if I understand correctly --


 * 1) As a set of statements in SECOND-ORDER logic, where the induction axiom takes the usual set-theoretic form of "If a set A satisfies blah, blah, blah, then A = N". THIS is what is usually meant by "Peano axioms".


 * 1) As an infinite set of statements in FIRST-ORDER logic, where the single induction axiom is replaced by an infinite schema of first-order statements. This axiom schema is also sometimes called "Peano axioms", but most of the time, people mean the former by this term, not this.

In any case, whatever you call them, they're not the same. One is a finite set of statements in 2nd-order logic, the other an infinite set of statements in first-order logic. The fact that the former "disallows" natural numbers other than 0, 1, 2, ... and forces a Peano structure to be unique up to isomorphism, and the fact that the latter "allows" nonstandard natural numbers other than 0, 1, 2, .... and allows infinitely many non-isomorphic models of the axioms isn't a contradiction. One simply is not the other.

Revolver 11:54, 18 Jun 2004 (UTC)

Does place holder mean acceptance as number?
Does the use of the numeral zero as a placeholder really imply an acknowledgement and true understanding of the concept of the number zero?? I'm not sure...my first inclination is to say "no". Revolver 07:05, 2 Sep 2004 (UTC)
 * The answer is most certainly no. The "invention" of the zero as a placeholder preceded the "discovery" of zero as a number by many centuries. Paul August 02:29, Nov 9, 2004 (UTC)
 * For a good article on the history of zero, see: Paul August 02:29, Nov 9, 2004 (UTC)

Circular
Integers are currently defined in terms of natural numbers, and vice-versa. 24.91.43.225 17:18, 14 Jun 2005 (UTC)

Identity
I am always confused why number theorists do not include 0 in N, since without it, there is no additive identity. Though it was not "discovered" until numbers had been in use for a long time, I think zero is quite "natural" for counting: I can imagine having 1, 2, or 0 objects, while -3 or 2 1/2 require some kind of logical "leap." I guess this explains why set theorists like 0 in N, since the cardinality of a finite set is always in N.
 * I don't know why number theorists don't include 0 (granting that they don't; I haven't talked to one about it recently). It's just their convention, I suppose, and doesn't really need a justification. All we can really do in an encyclopedia is report both conventions, which we correctly do, and I would like to suggest that detailed discussion of it take place in some other forum, because it's not adding value to the article to keep rehashing it. --Trovatore 17:03, 1 November 2005 (UTC)

Good point, I just couldn't restrain myself from jumping into such an interesting (though ultimately pointless) discussion :) Enough on the subject.  198.160.96.7 15:21, 2 November 2005 (UTC)

Reference to Frege-Russell definition added to "Other constructions"
I added a description of the Frege-Russell definition of the natural numbers (which works in New Foundations and related systems) to the "Other constructions" section.

Randall Holmes 21:33, 19 December 2005 (UTC)

Set theoretic definition
The set theoretic definition just added is incomprehensible as it stands. It is in severe need of definition of its components. What is a(b=c')? Also from its context, the first time is looks like it is an element of a set, the second time a set itself. Please elucidate. &minus;Woodstone 07:04, 26 June 2006 (UTC)


 * I've taken it out, for now. If anyone wants to readd, I would ask them please to explain the section, and maybe post it on the talk page first.


 * I'm particularly surprised that the definition given for "natural number" appeared to be a first-order formula. AIUI, if there is an infinite model for one of those, there are arbitrarily large models, so the natural numbers wouldn't necessarily be countable ...


 * RandomP 15:50, 26 June 2006 (UTC)
 * The natural numbers are definable by a first-order formula in the language of set theory. This does not in any way contradict the fact you cite, that any first-order theory with infinite models has arbitrarily large models.
 * Now I think it's also true (though I don't see that it follows immediately from the above fact; it probably follows more from the proof) that, for any infinite &kappa;, there are models of set theory with &kappa; things that the model thinks are natural numbers. But remember that just because the model thinks they're naturals doesn't mean they are. Some of them will be "fake" natural numbers, and will not satisfy the first-order definition in the real, Platonistic universe of sets. --Trovatore 05:15, 27 June 2006 (UTC)

Some people at Talk:Axiom of infinity were complaining that they did not know how to separate out the subset of natural numbers from the set whose existence is given by the axiom. I was offering a definition which avoids recursion and also avoids taking an "intersection of all sets containing 0 which are closed under the successor function.". Stripping out the formula, I said "In Zermelo–Fraenkel set theory, n is a natural number means ... it is either zero or a successor and each of its elements is either zero or a successor of another of its elements. Where zero is the empty set; and the successor of y is the set containing y together with all of the elements of y.". This can be proven to be equivalent to the definition given earlier in the article. JRSpriggs 11:19, 27 June 2006 (UTC)

So everyone's agreed that whatever is defined by that formula isn't the set of natural numbers (but every set fulfilling the formula is a superset of them)?

RandomP 13:06, 27 June 2006 (UTC)


 * I think you may have misinterpreted what I wrote. I'm not sure which exact formula you're referring to, but there certainly is a first-order formula that precisely defines the set of natural numbers. Some models will interpret the formula incorrectly, and therefore may think some things are natural numbers that in fact are not, but that doesn't change the fact that the formula is a correct definition --Trovatore 16:05, 27 June 2006 (UTC)


 * Quite probably. If you think a different definition should be in the article, don't let me keep you from adding it &mdash; it's just that what I removed did indeed seem ununderstandable, and possibly incorrect.
 * RandomP 16:21, 27 June 2006 (UTC)

To RandomP: No, my definition was correct. The prime symbol $$y\prime$$ is another notation for the successor of y, it is the same as S(y) used in the article. In ZFC set theory, every thing (including 57 or any other natural number) is a set. The definition used in the article says that a natural number is a member of the intersection of all sets, U, which contain 0 and are closed under the successor operation. Suppose that n is either zero or a successor and each of its elements is either zero or the successor of another of its elements. Then I claim that n is an element of U. Suppose it was not. Then n could not be zero, since zero is in U. So n is a successor; say the successor of m. Then m is an element of n. If m were in U, then n would be also. So m is not in U and thus an element of n is not in U. Form the set W of elements of n which are not elements of U. W is not empty because it contains m. So by the axiom of regularity, W contains an element k which is disjoint from W. k is in W and thus in n, so k is either zero or the successor of another element of n, call it j. k cannot be zero because zero is in U and thus not in W as k is. Since j is not in W but it is in n, then j must be in U. So the successor of j is in U, but that means that k which is that successor is both in U and not in U, a contradiction. Thus the supposition that n was not in U must be false. Thus n is in U for any U which contains zero and is closed under successor. Thus n is a natural number. Thus my characterization of natural numbers is correct. JRSpriggs 05:46, 28 June 2006 (UTC)


 * In my previous message, I showed that a natural number as I defined it is a natural number as defined in the article. Now, I show the converse, that a natural number as defined in the article is a natural number as I defined it. Zero is clearly a natural number by either definition. If n is a natural number as I defined it, then I claim that its successor is also. If so, then the set of natural numbers as I defined them, call it J (if the axiom of infinity holds (as the article's definition assumes), then J can be obtained by applying the axiom of separation to the natural numbers as defined in the article), is one of the sets which contains zero and and is closed under successor. So when you take the intersection of all such sets to form the set of natural numbers as defined in the article you get a subset of J. This is what was to be proved. All that remains is to show that the successor of n is in J. Call that successor p. Since p is the successor of n, p is either zero or a successor. Now, consider any x which is an element of p. Either x is n or it is an element of n because that is what successor means. If x is n, then x is either zero or a successor because that is true of n. If a successor of w, then w is an element of x = n and thus of p. On the other hand, if x is an element of n, then x is either zero or the successor of w for some w in n. But if w is in n, then w is in p. So we have shown that, in either case, every element of p is either zero or a successor of an element of p. So p is a natural number as I defined it, i.e. p is in J. QED. JRSpriggs 04:33, 29 June 2006 (UTC)
 * Furthermore, my definition is superior to the definition in the article because my definition only requires the axiom of extensionality, the axiom of regularity, and the axiom of separation to ensure that it selects the right sets as natural numbers; while the definition in the article requires the axiom of infinity as well as those axioms to ensure that it selects the right sets. For example, suppose one was using V&omega; as a model of ZFC minus infinity. Then my definition would still work. But the definition in the article would fail because no set, U, containing zero and closed under successor would exist in the model. Consequently, a "natural number" defined as belonging to all such sets would include every set in the model, most of which are not really natural numbers. JRSpriggs 04:46, 29 June 2006 (UTC)

My definition of natural number derives from the fact that they are the finite ordinals. An ordinal is finite iff it neither is a limit ordinal (like ω) nor contains a limit ordinal. However, a limit ordinal is simply an ordinal which is neither zero nor a successor. So to be a finite ordinal (i.e. natural number) means that it is an ordinal which is either zero or a successor and contains only ordinals which are zero or successors. By the trick of requiring that the elements of n which are successors are successors of elements of n, we can force n to be an ordinal. That is how I arrived at my definition. Now, if no one is objecting any longer, I will re-add my definition in a day or two. JRSpriggs 05:43, 30 June 2006 (UTC)

On rereading the section, I indeed don't see any mistakes. I'm assuming for now that I misread it the first time, but I also must wonder whether it couldn't be phrased differently and be less confusing; however, no mathematical objections to it.

My apologies for that.

However, I also suggest the information go into the "the standard construction" section, and use its terminology; my edit summary is still correct in that it's redundant (once you read it properly) with the information in that section, and some editing might be required to make that clear. That all can be fixed once it's back, though. If you believe it's useful information, put it back, and the merciless editing can happen afterwards.

Sorry, again, for the holdup, and thank you for taking so much time to explain things.

RandomP 08:39, 30 June 2006 (UTC)
 * Done. Thanks for reconsidering. JRSpriggs 04:57, 1 July 2006 (UTC)
 * Another way to look at this is that the old definition was saying that the set of natural numbers is the set of numbers which you can count up to. While I was saying that a natural number is something which one can count down from to zero. JRSpriggs 09:29, 2 July 2006 (UTC)

Tweaking the standard construction

 * The old definition is also the one overwhelmingly used; while your definition is a nice reformulation, and doesn't read too much like WP:OR, the old definition is definitely the one that should be given more prominence.
 * I'm unhappy with this edit, in particular.
 * If I am (again :-) ) wrong about this, and your definition enjoys significant popularity, please include some references?
 * It seems to me you're replacing the standard definition with one you like better, and one that cunningly misses the main point (though I must admit Wikipedia naming conventions help you with this): There is a set of natural numbers, and it is the existence of this set that is required for such essential thing as defining addition, or even the successor function (the "function" defined here technically isn't one, as its domain is the proper class of all sets).
 * The current article glosses over this, as well, which is unacceptable.
 * RandomP 11:58, 2 July 2006 (UTC)

Does a natural number, such as 4 = {{Who},,{{Who},},{{Who},,{{Who},}}}, cease to be a natural number, if it is located in a structure which lacks ω or perhaps even larger natural numbers, like googolplex? Does a non-natural number become a natural number because it is in such a structure? In other words, is being a natural number context dependent? (Of course, it depends on its elements being present and the element relation among them being unchanged.) That is my concern. JRSpriggs 05:37, 3 July 2006 (UTC)


 * In order (and with many caveats, as I'm somewhat sleep-deprived): Yes, it ceases to be a natural number, for some people;  the second question does not make sense to me, which is probably my fault;  maybe:  properties of the natural numbers depend on whether the set of natural numbers exists (and the Peano axioms make use of the set of natural numbers).


 * Some justification:
 * you can find uncountable models of the naturals
 * Goodstein's theorem is true for natural numbers, but fails in some models of Peano arithmetic.


 * The last point, in particular, is the main issue: not Goodstein sequences specifically, but that there are theorems about the natural numbers that Peano arithmetic alone cannot prove.


 * RandomP 06:04, 3 July 2006 (UTC)

Formal Definition
I don't understand why there's even an argument. If you look inside any decent college math textbook that defines the natural numbers, you will see a definition that says, "the set of natural numbers is the union of all inductive sets." Your arguments about including 0 is pointless because that's not what the natural numbers are about. The most important thing about the natural numbers is that it's an inductive set. Moreover, it's THE inductive set. —The preceding unsigned comment was added by Davexia (talk • contribs) 21:58, 7 July 2006   (UTC)


 * There is already a section on formal defintion in the article. I think that's enough and there is no need to add in the formal definition in the intro. Also, is it correct that


 * the set of natural numbers is the union of all inductive sets


 * as you state? By the way, I found mathworld's iductive set clearer than our inductive set. Oleg Alexandrov (talk) 00:04, 8 July 2006 (UTC)
 * Our inductive set is about a concept from descriptive set theory. The article corresponding to the Mathworld article is at inductive set (axiom of infinity). The latter concept is not one that really needs to be named, and I don't think it ordinarily is named; people just talk about "a set closed under successor". --Trovatore 00:33, 8 July 2006 (UTC)
 * OK, then it is wrong to say that "the set of natural numbers is the union of all inductive sets". Rather, N is the smallest set which is closed under suscessor, up to isomorphism. Oleg Alexandrov (talk) 02:18, 8 July 2006 (UTC)

The standard definition (not my improvement) refers to the intersection (not the union) of inductive sets. But you must define what you mean by an inductive set. The definition which is appropriate in this case is a set which contains zero and is closed under the successor function. JRSpriggs 02:30, 8 July 2006 (UTC)

I'm not familiar with this description of N in terms of inductive sets. Is it standard? I note that another definition for N could be "initial object in the category of pointed unary systems". I like that description a lot, and I suspect it's equivalent to this notion of inductive sets, but I don't think it should go in the intro of the article. -lethe talk [ +] 03:44, 8 July 2006 (UTC)


 * "Inductive set" sounds like it should have a definite meaning, but I am not aware of any standard definition. Sometimes "hereditary set" is used for the same kind of thing, but that also has multiple meanings. JRSpriggs 04:58, 8 July 2006 (UTC)

Confused
I am confused by the definition
 * $$\sigma(A)$$ (for any set A) as $$\{x \cup \{y\} \mid x \in A \wedge y \not\in x\}$$

Should that not be more like:
 * $$\sigma(A)$$ (for any set A) as $$\{\{x \cup \{y\} \mid x \in A \} \mid \forall y \not\in x\}$$

As it stands it does not appear to define a set of sets. My maths is a bit rusty, so please comment. &minus;Woodstone 12:22, 8 July 2006 (UTC)


 * The formula in the article would be a correct way of defining successor that one could use, if there were a set of all n-element sets which we could call "n". (&sigma; is being used here for the successor function.) If A were "n" so defined, then x would be an n-element set and y would be a potential n+1st element to be added to the set. Does it make sense now? JRSpriggs 08:00, 9 July 2006 (UTC)

In the article n is defined as the set of all sets with n elements. So it is not enough to add one element to each of those sets. You would need to add all possible extra elements (in turn) to each of them. &minus;Woodstone 09:05, 9 July 2006 (UTC)


 * Assuming that A IS the set of all n-element sets, then $$\sigma(A)$$ as defined in the article is the set of all n+1-element sets. Perhaps you do not understand the notation. In the expression $$\{x \cup \{y\} \mid x \in A \wedge y \not\in x\}$$, the part on the left, $$x \cup \{y\}$$, is to be included in the set for EVERY x and EVERY y that statisfy the condition on the right, $$x \in A \wedge y \not\in x$$. Does that help? JRSpriggs 09:25, 9 July 2006 (UTC)

It does not state explicitly for every y. It might be interpreted as just any y. That's why I think the universal quantifier belongs in the expression. (And perhaps for x too). &minus;Woodstone 09:35, 9 July 2006 (UTC)


 * You cannot correctly include such quantifiers in this notation for sets using braces because you cannot apply the scope of the quantifier to the correct stuff. If you want to be so explicit, then you must abandon the braces and write down a formula such as $$\forall A \forall s(s \in \sigma(A) \iff \exist x \exist y[s = (x \cup \{y\}) \land x \in A \land y \notin x])$$. This implies that $$\forall A \forall x \forall y((x \cup \{y\}) \in \sigma(A) \Leftarrow [x \in A \land y \notin x])$$. JRSpriggs 01:40, 10 July 2006 (UTC)

Disclaimer
This article appears in need of improvement both in form and in content. I touched it, barely, to clean up a little formatting, but have not given it anywhere near the attention it should get. In particular, I am explicitly not vouching for the contents, which may be gibberish or genius (though I doubt either extreme). I'd also consider adding a mention of "natural number objects" in category theory. --KSmrqT 04:56, 9 July 2006 (UTC)

Countable sets
I have a question elicited by the fargoing abstractions in the article. With all this abstaction it looks to me like any countable set is defined to be the natural numbers. That would imply that the rational numbers are equal to the natural numbers. Correct or not? If so: didn't you go too far? If not: why are they different? &minus;Woodstone 09:46, 9 July 2006 (UTC)


 * Clearly, using a non-standard model of the natural numbers would cause a great deal of confusion so it should be avoided whenever possible. If none-the-less you use a non-standard model, then you must specify not just the base set (which could be any countable set as you said), but also which element plays the role of zero and what function plays the role of successor. With such additional information, the rational numbers used as the base set for a model of the natural numbers would be distinguishible from the rational numbers regarded as usual, as the smallest field extending the integers. JRSpriggs 10:14, 9 July 2006 (UTC)

The zero and successor would just be defined by the mapping onto the naturals (which exists by definition for a countable set). In the specific case e.g. using the "diagonal" count (skipping reducible fractions and alternating postive and negative). Without addition defined this does not seem distinguishable from the natural numbers as defined in the abstract sense. (Of course the ordering is different). &minus;Woodstone 10:32, 9 July 2006 (UTC)


 * So what is your point? That you can devise something which is confusing? And then be confused by it? JRSpriggs 01:45, 10 July 2006 (UTC)


 * Yeah, the rationals are in bijection with the naturals. The next question is: so what?  Remember that the naturals and the rationals are both monoids, and that they are not isomorphic as monoids.  They are both totally ordered, but not isomorphic as ordered sets.  It's only when you forget about all extra structures that you're allowed to view them as the same.  -lethe talk [ +] 02:23, 10 July 2006 (UTC)

Lack of references
I added the tag denoting lack of references, because I do think the article is poorly referenced. Since when could a set have been two different sets simultaneously, as is claimed in the first sentence? Besides, this claim is put to questional light in the very next sentence, where it is said that the numbers can be used for ordering: you cannot say "0th largest city in the country", can you? Next it is said that "Historically, the precise mathematical definition of the natural numbers developed with some difficulty. The Peano postulates state conditions that any successful definition must satisfy." and the first condition is that "There is a natural number 0", so why not to include zero? I think there's some great risk of misunderstanding.

Norman L. Biggs defines natural numbers in his book Discrete Mathematics using 11 axioms. From these axioms it easily follows that zero is not a natural number. I believe zero is usually included within the natural numbers only when the writer is not willing to state the axioms that define the set. Sure, you can construct axioms that generate all natural numbers and the zero, and I'm not saying that it is a big mistake to include zero, but the axioms defining the set should clearly be stated.

One part where I added the "" tag was "Arguably the oldest set-theoretic definition of the natural numbers is the definition commonly ascribed to Frege and Russell under which each concrete natural number n is defined as the set of all sets with n elements." Doesn't this claim need a reference, then? ZeroOne ( talk | @ ) 04:32, 11 October 2006 (UTC)


 * Thanks. I put back some of the fact at things that you find dubious. I had removed all of those earlier because it appeared to me that you were picking at too many things. For example, there is no need to cite a book in the sentence which claims that number theory people would want naturals to start with 1 and the computer science want them to start with 0. But you do have a point. Some statements in the article are in need references. Oleg Alexandrov (talk) 15:00, 11 October 2006 (UTC)


 * There could surely be some further references, especially to the historical section. However, at least one of the problems you identify IMO has more to do with the trouble to make mathematical semantics clear to readers who are not (yet) very advanced in maths.  You write Since when could a set have been two different sets simultaneously, as is claimed in the first sentence?  Now, I assume you refer to the sentence In mathematics, a natural number is either a positive integer (1, 2, 3, 4, ...) or a non-negative integer (0, 1, 2, 3, 4, ...).  No set is mentioned explicitly in this sentence; but we could use good definitions in forming sets.  Thus I interpret your objection in this way: You mean that as a consequence of this sentence, we would have the equalities
 * $$ \{1, 2, 3, 4, \ldots\} = \{\hbox{Natural numbers}\} = \{0, 1, 2, 3, 4, \ldots\}\,$$.
 * Now, as I hope you agree, this is not the intended meaning of the sentence. Rather, it is this:  As with all specific mathematical terms, in mathematics natural number has no other meaning than the meaning mathematicians agree to give it.  In this case, different mathematicians choose to give the term different meanings. Namely, some let it mean 1, 2, 3, 4,... and others let it mean 0, 1, 2, 3, 4,... .
 * In other words, there is no such thing as the set of all natural numbers, independent from the choice of meaning of natural number.


 * In my experience, it is virtually impossible to write a sentence with mathematical content that no single student can misunderstand; but if many misunderstand the text in the article, then it should be rewritten. If they misunderstand because they have a 'Platonic' view of terms, believing them to stand for 'the thing in itself' rather than for meanings more or less arbitrarily given by agreements and conventions, then perhaps we should write a main article on mathematical terminology and refer to it.  However, I doubt that preserving the text but adding exterior references (about who is using which convention)  will help any reader who believes that the sentence implies that two different sets are equal. JoergenB


 * As for history and logical arguments: they point in both directions, too. The notation N, Z, Q, R, and C was introduced by the Bourbakists.  They included 0 in N, which probably is a reason e.g. many algebraists (including me) do.  I thought this worth mention rather early; but another user didn't think so, and I do not plan to enter any edit wars.  Norman Biggs does not include 0.  Judging from the organisation of his well-known text-book in discrete mathematics, he may be motivated by the intrinsic logic argument.  (If you want the cancellation law $$ac = bc \Rightarrow a=b\,$$ to hold in full generality, you cannot risk someone picking c = 0.  On the other hand, if you want a 'neutral element for addition', you include 0.)  I personally know that this is the reason a Swedish author (Lars Nystedt) choose not to include 0 in a book intended for the education of future math teachers.  (I asked him.  The thing was a bit surprising, since the central school authorities in Sweden have decided that in Swedish scools 0 indeed shall be considered as a natural number and a member of N.)  If you go just a little further back, to the 'pre-Bourbaki world', you find rather large deviations from modern usage.  E.g. in the Concise Oxford English Dictionary, whole numbers and integers were considered as synonyms, and both meant the numbers 1, 2, 3, 4,... (and no others).
 * Another example (which indeed is in our reference list) is Edmund Landau: Grundlagen der Analysis (Leipzig, 1930). This is a very nice exposition of the construction of numbers step by step, from the Peano axioms for natural numbers and up to and including complex numbers.  Landau does not include 0  in the natural numbers; but he also defines all rational numbers as being positive.  For Landau, a rational number is whole (ganz), if it may be represented by $$\frac{x}{1}$$ for some natural number x.  He proves that the set of the whole numbers fulfils Peano's axioms, and that thus they may be identified with the natural numbers.  He doesn't use any notation for the set of all natural or rational or real or complex numbers, in either his or Bourbaki's sense.  (At least not in the original German edition; I haven't read the English translation.)
 * In a global encyclopædia, such older uses probably should be mentioned; but we do need to present common usage to the readers, and also variations in this. There could be good reason to include some stuff from wikis in other languages.  Some general references could be added at the end of the whole article (or separate sections); but I do not think there is much use in adding references for each sentence in the way ZeroOne seemed to suggest. JoergenB 17:03, 11 October 2006 (UTC)

Merge whole number here
The whole number article is nothing but an extended dictionary definition and has no good rationale for existing as a separate article. A brief note in the natural number article, stating that the term "whole number" is sometimes used for the naturals including zero, should be sufficient. --Trovatore 01:47, 29 October 2006 (UTC)
 * I agree. Paul August &#9742; 02:09, 29 October 2006 (UTC)
 * So pratically, natural numbers are any whole numbers. —The preceding unsigned comment was added by Niceguys (talk • contribs) 16:41, 8 November 2006   (UTC)


 * Don't you think that "whole number" might equally well be any "integer number"? &minus;Woodstone 17:35, 8 November 2006 (UTC)
 * Well, it's not a term used much by mathematicians, with any meaning. Those of us whose primary and secondary math classes used Houghton–Mifflin books learned that the whole numbers were the nonnegative integers, but nowadays I prefer to call those the naturals. --Trovatore 20:13, 8 November 2006 (UTC)
 * NRICH says that it can be positive or negative. Looking at whole number, it seems clear that it is an ambiguous term. So I think we have to turn whole number to a disambiguation page, or delete it. I agree that we shouldn't have a separate article on it. -- Jitse Niesen (talk) 23:32, 8 November 2006 (UTC)
 * Fair enough; a dab page seems fine. --Trovatore 00:10, 9 November 2006 (UTC)

Frege
What Frege defined in "Die Grundlagen der Arithmetik" were not natural numbers but cardinal numbers as one figures out by reflecting -- for example -- upon the set of all sets with #$$\mathbb{N}$$ elements. Such a set meets the definition but sure is no natural. —The preceding unsigned comment was added by Special:Contributions/ (talk)

Assessment comment
Substituted at 21:45, 3 May 2016 (UTC)