Talk:Natural number/Archive 1

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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)

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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 u_n+1 = u_n + 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. [1]

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:

1. 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).
2. 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.
3. 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.
4. 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.
5. 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 ≥ 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

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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).

• 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 ≠ ∞.

--Elektron 15:08, 2004 Jun 12 (UTC)

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)

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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 > or <. "No other natural numbers exist" is an icky way to disallow these. If you do allow use of < in the axioms, then you can say "there is no natural number a which satisfies n < a < 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, {+,>,=} in the axioms. --Elektron 07:13, 2004 Jun 18 (UTC)

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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 > or <.

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. 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 < 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" <, >, 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 < a < 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)

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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: [2] 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. −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 κ, there are models of set theory with κ 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 — 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_ω 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. −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". (σ 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+1^st 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. −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). −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 $\forall A\forall s(s\in \sigma (A)\iff \exists x\exists y[s=(x\cup \{y\})\land x\in A\land y\notin x])$. 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. --KSmrq^T 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? −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). −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 ☎ 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"? −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. [3] 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 195.176.59.181 (talk) 09:01, August 20, 2007 (UTC)

Assessment comment

The comment(s) below were originally left at Talk:Natural number/Comments, and are posted here for posterity. Following several discussions in past years, these subpages are now deprecated. The comments may be irrelevant or outdated; if so, please feel free to remove this section.

History is too short. Advanced material on generalisations needs to be moved from lead Tompw 12:12, 8 October 2006 (UTC)

Last edited at 22:26, 19 April 2007 (UTC). Substituted at 21:45, 3 May 2016 (UTC)

A successor

Maybe the Peano axiom "Every natural number a has a natural number successor, denoted by S(a)." should rather read "Every natural number a has exactly one natural number successor, denoted by S(a)." I know, that the last (mathematical induction) axiom implies this, but the proposed form (in contrast to the current form) is in accord with the use of the term successor here ("the successor").

Or change the use of the term successor so that it is not silently assumed that the successor is only one, could be even better. (something like a+1 instead of S(a) and "if a property is possessed by 'some' successor" instead of "... possessed by the successor" in the last axiom) --trosos 213.220.249.112 15:55, 13 January 2007 (UTC)

By the definition above, zero seems to be a natural number as it has a successor. Yet stretching this a bit further makes -1 a natural number as it too has a successor. I recall Peano got round the problem by first having an axiom defining zero as a natural number. Then the successor function is used to build the set and -1 never gets a look in. John H, Morgan (talk) 17:26, 2 December 2008 (UTC)

|N

I've seen the set |N discussed in the sci.math newsgroup occasionally; is this another notation for N, or does it have some other special meaning? Or is this an attempt to represent a special character within the limitations of ASCII? Should it be mentioned in the article? I found this informal definition in a newsgroup post:

The informal definition |N = {1,2,3,...} is usually taken to mean that |N is a set S such that

(1) 1 is a member of S

(2) for each member n of S, n+1 is also a member of S, and

(3) |N is a subset of every set, S, with properties (1) and (2).

I'm not clear on (3), except that I think it means that S can be any set containing consecutive naturals (and possibly other members as well), and therefore |N is a subset of any such set. — Loadmaster 23:13, 9 April 2007 (UTC)

It's an attempt to mimic blackboard bold in ASCII. It's almost always more confusion than it's worth. No, I don't think it deserves mention in the article. --Trovatore 23:16, 9 April 2007 (UTC)

Definition

Couldn't the first sentence just be "A natural number is a number greater than zero with no decimal separator"? For people who just want to know the basic definition without reading the entire article? —Preceding unsigned comment added by 86.76.137.45 (talk)

No. That is a definition by non-essentials. You should not confuse a number with a particular representation of that number (in this case the decimal representation). JRSpriggs 10:38, 22 April 2007 (UTC)

Peano axioms and isomorphism

The section on the Peano axioms claims that "All systems that satisfy these axioms are isomorphic". This would seem to contradict both the incompleteness theorem and the Löwenheim–Skolem theorem. 72.75.107.59 (talk) 01:29, 19 January 2008 (UTC)

Those refer to first-order logic. What the claim means is that all structures that satisfy the full Peano axioms, in the sense of second-order logic, are isomorphic. --Trovatore (talk) 01:33, 19 January 2008 (UTC)

Aren't the axioms listed there ("these axioms") all first-order? 72.75.107.59 (talk) 01:40, 19 January 2008 (UTC)

No, the full axiom of induction is not first-order. It becomes a first-order axiom schema if you limit the properties being considered to ones that can be defined by a first-order formula. --Trovatore (talk) 01:56, 19 January 2008 (UTC)

Number 4

this is topical: Mathematics to Retire Number 4--Billymac00 (talk) 14:58, 4 April 2008 (UTC)

This is funny, but I don't think it should be in the article. Oleg Alexandrov (talk) 15:24, 4 April 2008 (UTC)

THE Natural number

I am new to wiki and not sure if this is the proper way of posting a question, but I thought THE natural number was e (2.71828 18284 59045 23536...) since it appears in nature so often. (i.g. birthrates) Why is it not even noted on this page?

Just thought It should be noted since a single letter is hard to search for. see [4]

Fozforic (talk) 14:00, 23 September 2008 (UTC)

The term natural number, in mathematics, universally refers to the concept treated here (either the nonnegative integers or the positive integers, depending on your taste), and certainly does not include e, which however is the base of the natural logarithm function. The word natural appears in the names of both concepts, but that shouldn't be taken to indicate any close connection between them. This is in general the way mathematical nomenclature goes — multi-word terms mean what they're defined to mean, and their names should usually be taken as historical artifacts rather than as something you expect to be able to figure out the definition from. --Trovatore (talk) 17:02, 23 September 2008 (UTC)

This raises the question whether for the benefit of really ignorant people we should add a hat-note to disambiguate this, saying for example "For the base of the natural logarithms, see e (mathematical constant).". JRSpriggs (talk) 05:05, 24 September 2008 (UTC)

No, a search for e will get the reader to the desired page. --Salix alba 06:04, 24 September 2008 (UTC)

citation needed

i removed the citation needed on "Some authors who exclude zero from the naturals use the term whole numbers, denoted \mathbb{W}, for the set of nonnegative integers. Others use the notation \mathbb{P} for the positive integers." if a citation is needed for that than there is quite a bit else that needs citation in this article. these symbols are very often found in math textbooks.

I thought that $\mathbb {P}$ referred to the set of prime numbers….
― Kinkydarkbird (Talk Page) 09:23, 9 January 2009 (UTC)

i've seen it for both, and i have at least one text with me that uses $\mathbb {P}$ for the positive integers. —Preceding unsigned comment added by 71.192.103.225 (talk) 06:05, 26 January 2009 (UTC)

Natural numbers as sets

The article mentions the alternative encoding of natural numbers as sets by 0 = {} and n+1 = {n}. I have heard that this encoding was used by Peano. Does someone know the reference? --Jan 91.180.52.246 (talk) 22:25, 5 January 2009 (UTC)

That would have been an odd thing for Peano to do, since his axiomata begin with $\mathbb {N} _{1}$. I believe that the construction that you here mention was proposed by Russell. —SlamDiego_←T 06:04, 20 January 2009 (UTC)

Non-word?

Wow, somebody really needs to learn more English morphology. Regardless of whether “definitionally” was superfluous (I was trying to capture that for which some other editor had been reaching with the mistaken claim that the inclusion of zero were “explicit”), it's a perfectly proper English word. —SlamDiego_←T 06:00, 20 January 2009 (UTC)

0 is not the empty set

In the chapter "History of natural numbers and the status of zero" it says that 0 is the empty set. Sorry, but that is wrong. —Preceding unsigned comment added by 80.165.82.22 (talk) 19:49, 18 February 2009 (UTC)

What do you mean by "wrong"? There are multiple ways to define natural numbers in terms of set theory. Not all of them define 0 as the empty set, but the one that's generally considered "standard" for use in mathematics (whether you're doing mathematical logic or teaching undergraduates) very definitely does. And all of this is explained pretty clearly in the article. --75.36.134.30 (talk) 15:05, 26 February 2009 (UTC)

Redundancy?

A question for you all: should we delete the text "This set is countably infinite: it is infinite but countable by definition. This is also expressed by saying that the cardinal number of the set is aleph-null" in the Notation section? I tried editing it out for the following reasons:

• The content is repeated in the Generalizations section, so it is redundant.
• It goes against a suggestion in the Wikipedia Manual of Style for mathematics articles: "A general approach is to start simple, then move toward more abstract and technical statements as the article proceeds."

My deletion was reverted by another Wikipedian (see the history), who later gave the following explanation for the revert:

• Since "countably infinite" and "aleph-null" are defined in terms of the set of natural numbers, it seems appropriate to me to mention that fact in the section dealing with notations for the natural numbers and related things.
• A little redundancy helps communication.
• The section on generalization seems too far down in the article to introduce these ideas.

Any opinions on this? FactSpewer (talk) 03:27, 15 April 2009 (UTC)

Quaeler would you please choose your formulation and reedit the page?

You deleted

Which one we should use depends on which is better suited for our purpose. E.g. in number theory, only the positive integers are denoted by natural numbers whereas in algebra it is convenient to denote the non-negative integers as the natural numbers.

and below

Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense.

and commented in the history

18:54, 24 June 2009 Quaeler (talk | contribs) m (21,121 bytes) (Reverted good faith edits by Achim1999; Personalization of prose unencyclopedic; heavy mathematical drill down in opening paragraph makes the article less accessible to the average reader.. using [[WP:TW) (undo)

Well, I "personalization" with the words "we" and "our" because I thought this is good style! If you prefer you can use the passive form like

Which one should be used depends on which is better ....

And to your point "heavy mathematical drill down": You should immediateley delete the term "Ramsey theory" Argh! This is much heavier, why not using "combinatorics" ?

Moreover you (ther reader) could ask why this set is called "Natural" numbers? This never is done in this article! I open the eyes of the reader hopefully by stating ".... whether 0 is natural in the intuitive sense."

I hope in good faith you will take up these ideas and improve the article concerning these points and not only delete important(?) information. Regards, Achim1999 (talk) 19:33, 24 June 2009 (UTC)

I see by your talk page that you're no stranger to introducing verbiage that people have issue with; similarly, i found that your edits did nothing to improve the readability of the page and introduced no 'improvement'. Additionally, articles are not a place to wax philosophic as your text was doing. To be more to the point, i don't see that the article needs improvement in the areas which you attempted to introduce it and, as such, see no place to "take up these ideas and improve the article". Quaeler (talk) 20:52, 24 June 2009 (UTC)

The sentence "Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense." has no meaningful content and is stylistically inappropriate for an encyclopedia. JRSpriggs (talk) 20:56, 24 June 2009 (UTC)

Are you pulling my leg? 2) The sentence "Of cource we can discuss or philosophize endless whether 0 is natural in the intuitive sense." should be taken, to tell the reader, somewhere in the main article if you like, WHY is this set of numbers called NATURAL numbers! 1) The sentence "Which one we should use depends on which is better suited for our purpose. E.g. in number theory, only the positive integers are denoted by natural numbers whereas in algebra it is convenient to denote the non-negative integers as the natural numbers." (you may reformulate it in passive form, if you dislike the we/our style) gives the reader an explanation why there are two different sets used. And therefore exactly this is the position to place it, IMHO. And this should be undoubtfully an improvement. Please judge because on the matters concerning the facts and not because your personal feelings are influenced by reading e.g. my talk-page or because you know I favour to behind-ask critically given states, if you feel to act like as a reviewer. :-( Regards, Achim1999 (talk) 21:17, 24 June 2009 (UTC)

To be pedantic, the order of my analysis was:

1. Notice new edits on this article because it's on my watchlist.
2. Reviewed the edits.
3. Thought to myself, 'wow - this shouldn't be in the article'
4. Rolled back the edits; due to my tools, this also opens your talk page.
5. Read your talk page.
6. Thought to myself, 'oh - that makes a little more sense'

and not:

1. Came across your talk page.
2. Read your talk page.
3. Thought to myself, 'i should go scrutinize his edits'.

So i have originally judged the matter based on solely on your work, and not the impression your talk page later propped up. Quaeler (talk) 00:22, 25 June 2009 (UTC)

It seems to me you favor meta-discussion to avoid discussion on contens like just happens. Sorry, therefore I have read your talk page and get the impression that you already are in the set of guys you are hunting for in the eyes of some other faithful WP-users. Therefore my advice: think twice about your article-editing-habit here on WP! BTW: I still hope that "Quaeler" is not the name/word in german for your intention here on WP. :-/ Regards, Achim1999 (talk) 10:04, 25 June 2009 (UTC)

About zero as natural number

In the article I read "To be unambiguous about whether zero is included or not, sometimes an index "0" is added ..." The funny thing is this isn't unambiguous as in my country (Belgium), zero is a natural number, and this symbol $\mathbb {N} _{0}$ means without zero, because $\mathbb {N}$ represents the natural numbers, which is with zero. Also for other number sets a sub-index 0 means "without zero" (in Belgium). Something extra I forgot first to mention: some people may wonder why things like $\mathbb {Z} ^{+}$ includes zero, but zero is in Belgium a positive AND a negative number so it is logic that in that case things like $\mathbb {Z} ^{+}$ means the positive numbers, which is with zero. In other countries in Europe is this not always the case, it depends.193.190.253.144 (talk) 21:47, 7 June 2008 (UTC)

Interesting -- it strikes me as really counterintuitive to write $\mathbb {N} _{0}$ for the specific purpose of excluding zero. However I also have not come across this notation as a way of allowing zero, which is what the article currently claims -- can anyone find this attested somewhere? --Trovatore (talk) 22:49, 7 June 2008 (UTC)

193's statement is at least partially confirmed by the Dutch Wikipedia. In the article Natuurlijk getal (Natural number) the notations $\mathbb {Z} ^{+}$ and $\mathbb {Z} ^{-}$ are given as including 0, while $\mathbb {Z} _{0}^{+}$ and $\mathbb {Z} _{0}^{-}$ are given as excluding 0. Also Positief getal (Positive number), Negatief getal (Negative number), and 0 (getal) (0 (number)) state that in Belgium the number 0 is considered both positive and negative. Dutch is one of the official languages of Belgium. The French Wikipedia is equivocal; for example, it defines Nombre positif (Positive number) as: un nombre qui est supérieur (supérieur ou égal) à zéro ("a number that is greater than (greater than or equal to) zero"), without explaining when the parenthesis is supposed to be in effect. Next thing, the French article states: Zéro est un nombre réel positif, et est un entier naturel. Lorsqu'un nombre est positif et non nul, il est dit strictement positif. ("Zero is a real positive number, and is a natural number. When a number is positive and non-zero, it is called strictly positive.") The Spanish Wikipedia, at Número positivo, has a similar text but is slightly clearer in expressing that the meaning is ambiguous. In Germany and Italy the number 0 is unequivocally neither positive nor negative.  --Lambiam 07:56, 8 June 2008 (UTC)

In French-speaking European countries, the word "positive" used to be used much as it is in English. "x is positive" meant x > 0. Now there are competing conventions. What used to be called positif ou nul (nonnegative) is now often positif (positive), and what used to be positif is now often strictement positif (strictly positive). (The word nul is an adjective that means "being zero".) I suspect that the change came about mostly because of the choices made by Bourbaki. The newer terminology is not universal, but it does predominate in France. Claims that the older usage has disappeared are probably exaggerations. Also, Canada has not followed French-speaking European countries with regard to the meaning of "positive". 128.32.238.145 (talk) 22:49, 16 November 2008 (UTC)

Since there is a lot of confusion in this area, I suggest we follow ISO so long as there is no reason to deviate from it. In particular, ISO 31-11 says that $\mathbb {N}$ includes 0, and also defines $\mathbb {N} ^{*}=\mathbb {N} \setminus \{0\}$. Normally I would have suggested to use $\mathbb {N} _{0}$ instead of $\mathbb {N}$, for clarity. I believe this is relatively standard practice in Germany. But as it seems this would only increase the confusion for Belgian and Dutch readers, it's probably best not to do that. --Hans Adler (talk) 11:10, 28 April 2009 (UTC)

There's an ISO for math??? That's terrible; that should not exist.

Of course I personally include zero in the naturals, but it's not because of any silly standards org that no one's ever heard of (in mathematics). I think the correct solution for the article is just to report that some include 0 and some don't, which is the truth. ISO should just be ignored, as is the practice of mathematicians generally. --Trovatore (talk) 18:07, 28 April 2009 (UTC)

I think such standards actually have the potential to be quite influential. What I learned at school seemed to be directly inspired by DIN 5473. I agree that such standards don't have much influence on university maths, and note my disclaimer about reasons to deviate from them. But keep in mind that most people who deal with concepts such as the natural numbers after school are engineers, for whom industry standards generally are relevant. Another point is that the committees behind such standards often do research that is similar to what we as Wikipedia editors are doing when we try to standardise our notation across articles: They try to find out which usage is more common, and they try to identify historical trends. Taken all this together, I think all else being equal we should prefer the usage prescribed by an ISO standard.

The DIN norms for maths, which no doubt influenced the ISO norms, are due to the Ausschuss für Einheiten und Formelgrößen, which was founded in 1907 by 10 scientific organisations including the Deutsche Physikalische Gesellschaft and the Verein Deutscher Ingenieure. [5] The mathematical norms function like a manual of style: If they manage to convince people or organisations, they will follow them. After reading our article ISO 31-11 I must say it seems to be perfectly sane. It would seem strange for me to not follow them simply to prove one's independence. Of course one problem with these standards is that the practice of only making them available for a lot of money is incompatible with scientific culture. --Hans Adler (talk) 20:17, 28 April 2009 (UTC)

Our standards should be the same as they always are — usage in the literature and the mathematical community, period. What some standards body claims is irrelevant. I have no specific gripe about any of the choices I saw in a brief scan; that isn't the point. Our job is to reflect the prevailing usage, and at the moment I believe it is still correct to say that both usages of natural number are current. Therefore we should say so. --Trovatore (talk) 20:44, 28 April 2009 (UTC)

I didn't want to say otherwise, but I see now how what I wrote can be misunderstood that way. I came here because anonymous editors at countable set have been removing the 0 from the natural numbers. (An ISO standard is probably a much more convincing argument for such users than a literature survey, but that's of course not relevant here.) However, I do think that we shouldn't have to define what $\mathbb {N}$ means in every single article that uses this symbol; we should have a general convention for this, and $\mathbb {N} /\mathbb {N} ^{*}$ is probably the best choice.

This discussion seems to have established (to my surprise) that $\mathbb {N} _{0}$ is ambiguous. Therefore it seems that the current text introducing the symbols needs changing. I am proposing to say that the natural numbers with/without 0 are denoted by $\mathbb {N} /\mathbb {N} ^{*}$, and that it is also common to denote them by $\mathbb {N} _{0}/\mathbb {N}$. If it can be sourced, I would also say that $\mathbb {N} _{0}$ is ambiguous because it can denote $\mathbb {N} ^{*}$. --Hans Adler (talk) 21:17, 28 April 2009 (UTC)

Since we seem to agree that in current use the term "natural number" has become ambiguous, so is necessarily the notation $\mathbb {N} .$ We should not try to hide that unfortunate fact. −Woodstone (talk) 22:39, 28 April 2009 (UTC)

I don't want to hide the ambiguity. But like every good textbook we can say, the symbol is ambiguous and this is how I use it. Only that we need to be a bit more subtle and can't say explicitly how we use the symbol, both for stylistic reasons and because we will never get all articles consistent with whatever convention we choose. When we use the symbol $\mathbb {N}$ in Wikipedia, then in many cases (probably the majority) it makes a difference whether 0 is included or not. See WP:WikiProject Mathematics/Conventions for how we have dealt with some similar problems. I believe that use of the symbol (but not of the term "natural number") needs to be standardised across Wikipedia. This talk page seems to be the best place to talk about this, although depending on the outcome of the discussion here it should of course be proposed on the Conventions page as well. Perhaps something like the following works:

A recent trend is to denote the natural numbers including 0 by $\mathbb {N}$ and the natural numbers without 0 by $\mathbb {N} ^{*}$. [6] This trend is reflected in recommendations for scientific writings such as ISO 31-11. However, the traditional practice of writing $\mathbb {N}$ for the natural numbers without 0 is still widespread. --Hans Adler (talk) 23:44, 28 April 2009 (UTC)

Yesterday evening, I stumpled about a paragraph in "Numbers" from Ebbinghaus et a., Springer, 1991, which states that 1stly R. Dedekind liked to start the Natural Numbers with one and 2ndly that G. Peano acknowledged that he was influenced by Dedekind's postulates when he defined his axioms for the Natural Numbers. Thus it appears to me that already in the beginning of the foundation of formal logic / axiomatization (around 1880) this "0 is / is not in N"-war started. ;-) Regards,Achim1999 (talk) 15:12, 8 July 2009 (UTC)

Formal Language

I know this isn't the Simple wiki, but even so, is it really necessary to say "Either a member of the set of the positive integers, or a member of the set of the non-negative integers", instead of leaving "sets" out of it and just saying "Either a positive integer or a non-negative integer"? 174.46.172.13 (talk) 10:35, 30 June 2009 (UTC)

I understand your point, and I tried to address it. Hans Adler 10:55, 30 June 2009 (UTC)

Do number theorists start the natural numbers with 1?

Although some number theory texts do start with 1, there are also many distinguished number theory texts that start with 0, and some of them are not so recent. For instance, see

• Serre, A course in arithmetic, Springer-Verlag, 1973, p. 115.
• Weil, Basic number theory, second edition, Springer, 1973, p. XIII.
• Ramakrishnan and Valenza, Fourier analysis on number fields, Springer-Verlag, 1999, p. 138.
• Rotman, Advanced modern algebra, Pearson, 2002, p.1. (This isn't really exclusively a number theory text, but the definition occurs in a section called "Some number theory".)

Therefore I suggest removing the bit in the history section about number theorists keeping the older tradition of starting with 1. --FactSpewer (talk) 20:44, 8 November 2008 (UTC)

I don't understand in what way Woodstone's rewrite of the opening paragraph is "more balanced". I feel that the original wording matches actual usage better, because it is true that currently "natural number" can mean either an element of {1,2,...} or an element of {0,1,2,...}. The first sentence of the revision suggests that the former is a thing of the past, which is not the case. The second sentence of the revision suggests that all authors in mathematical logic, set theory, and computer science start with 0 (I would hesitate to make such a claim), and is ambiguous on whether starting at 1 is done by everyone else, or by no one else, or ... . --FactSpewer (talk) 04:34, 28 April 2009 (UTC)

After a neutral "0 is in or not", the preceding version had a phrase: "the latter is especially preferred in (some fields)". So only a positive statement for including 0. I wanted to add something positive for not including 0. It cannot be doubted that 0 is a relatively new concept compared to the positive natural numbers. So "originally start from 1" is true. Some specific fields now often include 0, is true as well. I agree this is not universal in those fields, so we could add a remark about that, and make explicit that other fields stick to the original definition. −Woodstone (talk) 07:44, 28 April 2009 (UTC)

OK, thank you; I understand your thinking now. Probably whoever wrote especially preferred meant not that it was better, but that it was more common in those fields; but I agree with you that that wording could be construed as making a value judgment, so let's avoid it. I'll just word the opening sentence to make it clear that both conventions are used now; and I'll leave the history to the history section, which agrees with what you say, and expands upon it in detail. --FactSpewer (talk) 04:15, 12 May 2009 (UTC)

Divisibility and 0 bites each other very much in theory. :) Thus 0 was never paid attention in classical number theory and it was not missed! Achim1999 (talk) 18:55, 24 June 2009 (UTC)

Division works somewhat differently in number theory and analysis. In analysis one is simply not permitted to divide by zero. In number theory division is replaced by the notions of divisibility and congruence, where divisibility and zero get along just fine. The positive integers under the divides relation m|n form a distributive lattice that is not complete, whereas the nonnegative integers form a complete distributive lattice with 1 at the bottom and 0 at the top (the opposite of the usual convention for naming top and bottom in a lattice). In particular, without zero one can take the GCD of any infinite set of positive integers but not the LCM, whereas with zero one can take both the GCD and LCM of any infinite set of nonnegative integers. Hence number theory is better off when it includes zero because doing so expands the available operations.

In computer science the natural numbers always start from 0, as a consequence of the convention of breaking the cycle ... < 110 < 111 < 000 < 001 < 010 < ... between 111 and 000 when interpreting bit strings as unsigned integers, more precisely the integers mod 2ⁿ for Ironically computer keyboards break it between 000 and 001 (so to speak) as a result of typewriters having always done so, putting 0 at the right. Rotary telephones also did so, but that was because dialling n produced n pulses and if 0 had produced no pulses instead of ten pulses the switchboard would never hear 0. --Vaughan Pratt (talk) 05:40, 17 August 2009 (UTC)

Only mathematical well-defined statements wanted in the definition!

Sorry, do be pressed to open this section: But we dislike to see subjective, social unnecessary statments in the beginning (perhaps also in the wohle article?) of this scientific-supposed-to-be article! What N is, is stated precisely. No need for your personal oppinion to be added!

Regards, Achim1999 (talk) 11:49, 5 July 2009 (UTC)

I support the clarification there that the distinction is a matter of each author's convention. Without that clarification the "or" is more difficult to understand. — Carl (CBM · talk) 12:33, 5 July 2009 (UTC)

Achim, your argument makes no sense, and your attempt to enforce your removal of a stylistically necessary clarification by means of wiki-lawyering [7] was completely out of order. Given your obvious problems with the English language you should definitely not be edit-warring over style. You are probably not aware of it, but that's exactly what you are doing. Hans Adler 13:36, 5 July 2009 (UTC)

People who can not or want not to argue by contens but prefer to argue by pointing out wording, style and typos should be better ignored with their talk. :-(

Regards, Achim1999 (talk) 21:30, 5 July 2009 (UTC)

"..depending on context." is an empty phrase which is always correct. Thus adding this gives no further information but only blows up the writing. Escpecially we are here in a mathematical definition, hence one expects short, simple and clear wording, and no story-writers using fill-words. Argh!

Regards, Achim1999 (talk) 21:30, 5 July 2009 (UTC)

In dingo culture, a human being is either a man from asia or a woman from africa. What is unclear? And what becomes clearer by appending "depending on the context." to this sentence? *shaking my head* Regards, Achim1999 (talk) 21:35, 5 July 2009 (UTC)

Better wording-suggestion:

In mathematics, a natural number is either from the set {1, 2, 3, ...}, hence a positive integer, or from the set {0, 1, 2, ...}, then called a non-negative integer.

Regards, Achim1999 (talk) 21:58, 5 July 2009 (UTC)

That's not a good solution to the problem, and since your made-up example seems to confirm my suspicion that you are completely missing the issue, here it is: The purpose of "depending on context" or "depending on the author's convention" is to make it immediately clear that "either ... or" separates two different conventions rather than being part of a single, universally accepted definition of the natural numbers. This is an unusual situation, especially for such an elementary and well-known notion. We can expect that most readers 1) are not mathematicians, and 2) are familiar with only one convention and will be surprised to learn there are two. Without very clear hints they may well not understand this, and instead think that for some reason which they don't understand we are describing the convention they know in impenetrable mathematical jargon.

Note that your dingo example has exactly the same problem. Without "depending on context" it would be possible that a dingo can say "two human beings" to refer to a man from Asia and a woman from Africa. Adding "depending on context" or "depending on the dingo's convention" would clarify that there are two dingo dialects: one in which "human being" can only refer to a man from Asia, and one in which "human being" can only refer to woman from Africa.

Oh, and "depending on context" is of course not always correct. E.g. "an irreducible natural number is either the number 1 or a prime number" is correct, while "an irreducible natural number is either the number 1 or a prime number, depending on context" is plain wrong.

Your proposed reformulation is only marginally better in this respect than the version you have twice reverted to. It is also written in a very clumsy style, and any attempts to fix it would probably lead to what we have now (with or without the clarification).

By the way, I have two questions for you regarding "we dislike to see subjective, social unnecessary statments in the beginning":

1. Are you a single person or is "Achim1999" a group account? If you are a single person, who else do you think you are speaking for?
2. After my explanation, do you still believe that the explanation is unnecessary? If so, could you please elaborate what it is you don't like about it; I don't think there are many editors here who share your opinion.

Hans Adler 23:32, 5 July 2009 (UTC)

How about;

There are two conventions on what a natural number is:...? Septentrionalis PMAnderson 00:40, 6 July 2009 (UTC)

Thanks for finally give information of the contens and context which you think is ambigious. This appending "depending on the context." makes nothing clearer to me. Because this interpretation has nothing to do with english language, it occurs in german too and probably in many other language this is possible. But now I know what the misinterpretation could be. I only wonder if you think "Without very clear hints" is necessary why you are reluctant to give these very clear hints and make a better formulation which avoids exatly this supposed-to-be misunderstanding. E.g.:

In mathematics, there are two convention for a natural number: either it is a positive integer from the set {1, 2, 3, ...} or a non-negative integer from the set {0, 1, 2, ...}.

or perhaps

In mathematics, there are two convention for a natural number: either it is from the set {1, 2, 3, ...}, hence a positive integer, or from the set {0, 1, 2, ...}, then called a non-negative integer.

It should be easy, even for you ;-), to quickly generate contexts, such that "an irreducible natural number is either the number 1 or a prime number, depending on context." is a correct and useful statement. :)

To answer your many off-topic, non-mathematical social questions:

1. Achim1999 is no group-account. I even did not know that this is possible here.
2. I don't know. I have not made up my mind. But I was already pointed to this guidline(?) by other people / editors(?) here.
3. I never thought of your misinterpretation-possibility. I still have problems to realize how to interpretate this sentence as a single unique definition. Perhaps this is due to the fact, that one choice is a subset of the other.
4. I still don't know whether there is a clarification necessary, sorry. (see my answer 3).
5. I think appending "depending on context" makes nothing clearer -- I think you believe it should cause a grouping of some words in this sentence in your understanding. But this I can't realize, honestly.

Regards, Achim1999 (talk) 11:17, 6 July 2009 (UTC)

Group accounts are physically possible, in the sense that we cannot prevent someone else from sitting down with your password; they are forbidden, and grounds for blocking. Septentrionalis PMAnderson 16:32, 6 July 2009 (UTC)

BTW: I still wonder why you have pressed others (at least me) to make this discussion, and refuse to suggest a clear formulation which hits your point (two conventions are here in use) and therefore avoid such a waste of time. Regards, Achim1999 (talk) 11:41, 6 July 2009 (UTC)

I have pressed you? You reverted me twice for a reason that I suspect can be understood by nobody other than yourself ("we [sic!] dislike to see subjective, social unnecessary statments"), then you started the discussion here, and you unnecessarily notified me of it on my talk page. So far you have been reverted once by me, once (essentially) by JRSpriggs, and you have been told you are wrong by CBM. That's a score of 3 professional mathematicians and experienced Wikipedia editors disagreeing with you against a total of 1 editors (including you) agreeing with you. I think it's pretty clear at this stage that if you still don't understand things after I have taken the time to explain them to you, then it's entirely your problem. Wikipedia isn't school, and other editors here are not your teachers. We are not payed for explaining things to you that you don't want to understand. Hans Adler 15:05, 6 July 2009 (UTC)

Sadly, you seems really unwilling to stay factually. On such a base it makes no sense for me to discuss further! Sadly also, you like to judge wrong versus right by 3:1 opinnions getting in a few days. I can only hope that you act as a professional mathematician much more factually. Noone wanted you to act as a teacher -- at least I did not, but this seems also an off-topic attitude by you. Well, meanwhile other people took hand on this sentence and this very special interpretation-issue is gone. Unbelievable that you refused to make constructive changes which hit your point of (necessary?) clarification of those sentence. :-( Regards, Achim1999 (talk) 11:40, 7 July 2009 (UTC)

You were unable to express your concerns comprehensibly, which made it impossible for me to take them into account. Now the article has been changed to give slightly more detail. I think we had something very similar recently and somebody objected, but I don't have the time to look up the history now. In any case a first sentence with two parenthetical phrases is rubbish. And I have no idea why your strange objection doesn't apply to the present formulation.

If you have trouble communicating in English don't blame others for the fact. Thanks for the promise to end this silly discussion, by the way. Hans Adler 15:20, 7 July 2009 (UTC)

Yesterday a user added "... according to a more recent definition now also in common use." (I removed the parentheses). I'm really unhappy with the elastic wording "more recent". This should be made preciser. I believe it was first invented/defined by a formal approach in the 19th century to lay the foundation of numbers-notion (G. Frege / R. Dedekind). On the other hand the sentence should not become too long. Regards, Achim1999 (talk) 13:54, 8 July 2009 (UTC)

Keep wording short, clear and precise -- a main feature of good encyclopidae!

Well, just again this non-factual (I heard, there is a WP-guidline to generally not name people by calling their account names, especially if their action can be considered bad) liking author, disqualifing himself with this kind of attitude in the long run, sprang in action and triggers me to write this general comment regarding all WP-articles -- but I like to give two examples first to make my point hopefully clear:

Facts to this special example:

1. Since many months this section "algebraic properties" were here located without significant changes.
2. Some days ago, I noticed, that a,b, and c are in the mathematical interpretation so-called free variables. This I dislike, because in the langugual interpretation given by the context they should be considered to hold only natural numbers.
3. Therefore I added a very precise, short, mathematical statement as lead-in to all the other mathematical expressions in the following list which is in table-form and which is everything in this section.
4. Suddenly people spring in action, to avoid this mathematical clarification, which is also easily understandable by typcial to-be-expected non-mathematical readers, only to rephrase it with superfluous words like story-writes, using the interpretable word "properties" without any need! I did not expect others to act like very good mathematical writes, like say, D. E. Knuth, but thoughtless (in the most positive assumption) changes should be avoided which only blows up articles and even decrease readability.
5. Argh! :-( Sorry.

And a further example in this article which happend recently: Another guy/account disliked the word "computist". Even me, who probably had never heard this word, knew immeditely what the original text/author wants to express with this wording. Then this guy sprang into action who dislike this wording and must replace it by computer. But in common use this has today almost always a different meaning, thus this guy had to add an addtion to clarify this. Ironically he added in his comment to his change, that in pre-computer-age this word "computer" was used with a unique meaning of computist. But it now seems that he prefers fighting for the ambigusity of "computer" and supports it here by reusing it where totally unnecessary. My well reasoned revert, in fact I used his(!) reasons of usage in past to revert it, he reverted aagin but without any comments. Sorry, this I can not call faithful editing by heart. And to be frank, more the opposite creaps in my mind from learning other changes in other articles here in WP.

These kinds of attitude/behaviour of editing will surely not encourage more experts from special fields to sacrify their time to write WP-articles, a wish WP wants strongly.

BTW: Sorry, to sound in this writing to act like a well-sounded teacher. :-/

Regards, Achim1999 (talk) 10:48, 15 July 2009 (UTC)

No such guideline exists. Having to hunt through page histories to find out who you're talking about is very annoying, so giving names is almost always appropriate. Algebraist 10:52, 15 July 2009 (UTC)

I was told by a deserved editor on a user-talk page, that WP prefer to avoid explicit naming if criticizing editors at least. And it seems in retrospect that I criticized another deserved editor in that case. Maybe he meant only critizing of deserved editors and his wording looked to me more than a proposed policy. Sadly I have deleted all user-talk-pages (except my own) from my watch-list.

Regards, Achim1999 (talk) —Preceding undated comment added 11:19, 15 July 2009 (UTC).

You were told wrongly. Algebraist 11:22, 15 July 2009 (UTC)

Why do you think inline symbols are more readable than clear English text? Algebraist 11:49, 15 July 2009 (UTC)

Surely not generally, but you should think why they exist (resp. were invented). And in this case, section "Algebraic properties", they cause much better readability than prose. Anyway, I think it is not worth to discuss with people like CBM · talk who acts before thinking.

Have fun to change this article in the future without my support. Regards, Achim1999 (talk) 17:26, 18 July 2009 (UTC)

Sometimes making a small edit draws attention of other people to lingering problems in a page. I'm not thrilled at all with the table in the "algebraic properties" section. I don't see any reason why we cannot use prose, perhaps a bulleted list. I'll see what I can do. — Carl (CBM · talk) 11:46, 15 July 2009 (UTC)

Yes, prose would be an improvement. Algebraist

I made some changes. I'm still not thrilled with the "algebraic properties" section being followed by the "properties" section; they should probably be merged. Also, there should probably be a section on the key property of natural numbers, which is that they support proofs by induction. There is currently just a link to mathematical induction buried in the article. — Carl (CBM · talk) 12:04, 15 July 2009 (UTC)

Conventions

Saying that starting at 1 is the traditional convention, and that starting at 0 is the convention following a formal definition in the 19th century, is not entirely accurate. The issue is more complicated than this, and is best left to the discussions later in the article, where it is amply explained. For example, Peano's original definition started at 1, not 0 (it is only that some logicians modified his definition later on to start at 0, out of convenience). And both definitions occur in the modern literature. --WardenWalk (talk) 19:20, 1 August 2009 (UTC)

Could someone with more experience of these things than I (or access to a good academic library) please add a reference to a work which treats 0 as a natural number? The article makes it sound as though this is the more common modern day convention, but my 2nd edition copy of the Penguin Dictionary of Mathematics (for example) only lists the ℤ>0 version (which it also identifies as ℤ⁺, to further muddy the waters). Thanks. Aoeuidhtns (talk) —Preceding undated comment added 12:46, 5 February 2010 (UTC).

Literally any book on set theory or mathematical logic. I checked Levy, Jech, and Kunen's set theory books just now, and also Shoenfield; Hinman; and Boolos, Boolos and Jeffrey on the logic side. All of these begin the natural numbers at 0. I don't think I have seen any text in logic that begins the natural numbers with 1, although I can't say I've read every single book in the area. — Carl (CBM · talk) 13:26, 5 February 2010 (UTC)

I think some mention of convention by country should be mentioned. For instance here in the UK it is traditional to start the natural numbers at 1 whereas in France it is traditional to include 0. (I do not know of any references for this but it seems to be common knowledge amongst my professors and doctors (I'm doing a post grad in maths)). Porkbroth (talk) 20:08, 2 March 2010 (UTC)

I move to change the article to be in agreement with MathWorld, the standard online reference for mathematics. This would involve using "+" to indicate positive (instead of non-negative) and "*" to indicate non-negative. We shouldn't have the top 2 references that come up when you Google "natural numbers" disagree! Also, see: The MathWorld page on Positive Integers. watson (talk) 02:32, 1 October 2010 (UTC)

MathWorld is very very bad at terminology. We should not rely on it whatsoever. --Trovatore (talk) 02:41, 1 October 2010 (UTC)

Agreed. I don't see how mathworld is a standard reference for mathematics any more than wiki. If anything the opposite is true, mathworld being by and large the work of a single individual, whereas wiki is a cooperative effort involving a number of individuals. Tkuvho (talk) 04:36, 1 October 2010 (UTC)

Nominal numbers

I agree with Hans Adler that nominal numbers should be dealt with separately. Note to Fullmetalactor: I would say that the 23 in "I live on 23rd street" is functioning as an ordinal number (by the way, be careful when correcting spelling!) Ebony Jackson (talk) 07:29, 17 August 2009 (UTC)

Symbol unicode number

Would someone please mentions the symbol "ℕ" Unicode number next to it ? --DynV (talk) 07:16, 25 October 2009 (UTC)

You've made this request in at least two articles, maybe more, but I'm afraid I don't understand it. Could you please explain more clearly what you mean? --Trovatore (talk) 10:39, 25 October 2009 (UTC)

Certain symbols appear for some readers as a "ℕ" and for other readers as the intended symbol. When <math> is used, this problem does not occur. So in case of doubt, let us please use <math>. Bob.v.R (talk) 14:05, 25 October 2009 (UTC)

To Trovatore: Perhaps DynV means that he cannot read that character with his browser. It might look like an empty box or a question mark to him. (I presume he just cut and pasted the source to get it here.) He wants to know what numerical value is the code (the Unicode) for the symbol. JRSpriggs (talk) 19:43, 25 October 2009 (UTC)

That would be 2115 (hex), or entity "&#x2115;", showing as ℕ. −Woodstone (talk) 20:31, 25 October 2009 (UTC)