Is $0$ a natural number?

Question

Is there a consensus in the mathematical community, or some accepted authority, to determine whether zero should be classified as a natural number?

It seems as though formerly $0$ was considered in the set of natural numbers, but now it seems more common to see definitions saying that the natural numbers are precisely the positive integers.

It may be Italian education, but I've always been told, from 1st grade to 3rd year of my engineering degree course (present), that 0 ∈ ℕ, and never had any reason to believe the countrary. (We have ℕ₀=ℕ{0} when the need does arise.) — badp, Jul 21 '10 at 12:06
voted to close. The question is subjective, as is clearly indicated by the first sentence of the wikipedia article on Natural Numbers — Tom Stephens, Jul 23 '10 at 17:05
While definitely subjective, it might be the case that the asker genuinely does not know about the controversy and is in need of an answer to say "There is no answer". Whatever the case, I still voted to close. — Justin L., Jul 23 '10 at 20:49
@Justin, I know that there are mixed views (as indicated in the second paragraph of my question). But for the case of 1 being classified as a prime number, it seems the consensus view of the Mathematical community is that it should not count as a prime number. My actual question is 'Is there a consensus on whether zero is a natural number?' (although the question's title is simpler), so a suitable answer would be 'No, there is no consensus' combined with a quick demonstration from a few Mathematical dictionaries or articles that there are conflicting definitions. — bryn, Jul 24 '10 at 02:37
It is universally accepted that {1,2,3 .. } are Natural Numbers and that {0, 1, 2 ... } are Whole Numbers. Zero is not regarded a counting number as you do not usually count zero when you start counting. One sheep, Two sheep, Three sheep... Let's just stick with this so that there won't be any confusion. — Nick, Nov 13 '13 at 09:55
@Nick The responses to this question indicate that the definitions you propose are far from universally accepted. I agree that it would be great if everyone agreed on a standard, but I would argue strongly for the convention that 0 is a natural number. The convention $0\in\mathbb{N}$ doesn't mean you have to start counting at 0! — Alex Kruckman, Dec 02 '13 at 18:24
This question is not related to math, it is the consequence of an ambiguous notation which was used by Dedekind in 1888. To avoid ambiguity, ℕ* is used to exclude 0. I cannot manage to understand how this question is still not closed. — Wok, Mar 12 '14 at 14:35
In undergrad I had professors joke that 0 is a natural number if computers are natural to you (computer scientists usually have 0 as a natural number, but in a lot of math classes you don't have 0 as a natural number (though that's by no means a rule)). — Chris Rackauckas, Apr 02 '16 at 06:16
I have always seen that $\mathbb{N}$ is ${1, 2, 3, ...}$, while $\mathbb{N}_0$ is ${0} \bigcup \mathbb{N}$. — Awn, Nov 28 '17 at 11:53
I am taught that $0$ is not a natural number. The set of natural numbers is expressed thus: $$\mathbb{N} = {\text{natural numbers}} = {1, 2, 3, 4, 5,\ldots} = \mathbb{Z^+}$$ $0$ on the other hand is a whole number; a member (an element belonging to) the set of whole numbers. The set of whole numbers is expressed thus: $$\mathbb{W} = {\text{whole numbers}} = {0, 1, 2, 3, 4, 5,\ldots}$$ It is clear that $\mathbb{N}\subset \mathbb{W}$ namely because $0\notin \mathbb{N}$. However, @Awn is also correct, but pursuant to my Modern Mathematics 5 Fifth Year Book 1, $0$ is not natural. — Mr Pie, Jan 04 '18 at 05:40
Perhaps this might help $\longrightarrow$ https://www.studypug.com/au/au-year9/number-system-and-radicals/understanding-the-number-systems as it explains that $0$ is a whole number. — Mr Pie, Mar 19 '18 at 10:04
Perhaps this also might help ⟶ https://math.stackexchange.com/questions/2601711/is-this-a-valid-proof-that-there-are-infinitely-many-natural-numbers — Mr Pie, Mar 19 '18 at 10:04
For what it's worth: years ago when I was in school, Cambridge International (an international provider of exams and qualifications) defined $\mathbb{N}$ as ${0,1,2,3,\ldots}$, but these days they define $\mathbb{N}$ as ${1,2,3,\ldots}$. — ryang, Nov 15 '20 at 16:29
honestly analysts say you start at $1$, algebraists say you start at $0$ lol. — MyMathYourMath, Jan 20 '21 at 05:00
Of all the conventions that I have seen, I think that the use of $\mathbb N_0$ to mean the natural numbers without $0$ (whatever your convention) must be the most perplexing. I'll stick with my ugly but unambiguous $\mathbb Z_{\ge 0}$ and $\mathbb Z_{> 0}$. — LSpice, Apr 21 '22 at 15:33
As a student I had to accept both of these two contradictory definitions, but now I'm firmly in favour of including zero, and I think we do young students a disservice by teaching them the outdated definition of “natural number” as excluding zero. (If it weren't for the massive weight of history, I would be all in favour of re-jigging how we express ordinal numbers to correspond to “how many come before and not including”.) — Martin Kealey, Apr 25 '25 at 00:59

score 87 · Accepted Answer · edited Mar 08 '17 at 20:22

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Simple answer: sometimes yes, sometimes no, it's usually stated (or implied by notation). From the Wikipedia article:

In mathematics, there are two conventions for the set of natural numbers: it is either the set of positive integers $\{1, 2, 3, \dots\}$ according to the traditional definition; or the set of non-negative integers $\{0, 1, 2,\dots\}$ according to a definition first appearing in the nineteenth century.

Saying that, more often than not I've seen the natural numbers only representing the 'counting numbers' (i.e. excluding zero). This was the traditional historical definition, and makes more sense to me. Zero is in many ways the 'odd one out' - indeed, historically it was not discovered (described?) until some time after the natural numbers.

edited Mar 08 '17 at 20:22

answered Jul 21 '10 at 08:02

Noldorin

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I see plenty of both these days, but when I was at school and at university, I almost only saw them defined to be {0, 1, ..}. The elements of {1, 2, ..} were called the whole numbers in my school days. – Charles Stewart Jul 21 '10 at 09:06
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Maybe it's because I'm a student of physics that we do things slightly differently, but we seem to call only the counting numbers 'natural numbers'. 'Whole numbers' is just an informal way of describing all integers. – Noldorin Jul 21 '10 at 09:12
That definition of the whole numbers seems strange to me now, but back in my youth, that's how it was for me... – Charles Stewart Jul 21 '10 at 13:38
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Indeed, I can see why they were defined that way. There's a surprising lack of consistency in this area of naming. – Noldorin Jul 21 '10 at 13:50
Well, there is always a bourbaki definition, and it may be suitable to include one "unit élément". – awllower Apr 05 '11 at 10:18
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I would say that in number theory you will probably see $\mathbb N={1,2,\ldots}$, but in set-theoretical textbooks $0$ will be included as a natural number. (It is a natural approach in that contexts, since they're defined as finite ordinals. – Martin Sleziak Sep 17 '11 at 12:09
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Including 0 simplifies things. PA is just one example. – Kaveh Jul 28 '13 at 14:36
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@Kaveh: Not including $0$ simplifies other things. For example, you can then define rational numbers as (equivalence classes of) pairs of an integer and a natural number; no explicit exception for $0$ needed. And also the equivalence relation can then be easily stated by $(a,b) \equiv (ac,bc)$ for any $c\in\mathbb N$ (again, no exception needed). – celtschk Aug 01 '13 at 17:12
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@celtschk: Those things that you mention still occur with more general fraction fields derived from integral domains, in which you still have to distinguish between $0$ and non-zero elements. – user21820 Jun 03 '14 at 07:37
@Charles Stewart for me its totally opposite, natural numbers were always 1,2,3... and I have never heard about "whole numbers" before your comment. – Anixx Jan 08 '15 at 10:24
@Annix: I'm not sure why you wrote this comment. Are you claiming ignorance, authority, or something else altogether? – Charles Stewart Jan 09 '15 at 19:30
@celtschk If you define rational numbers that way, you miss the number 0 ;) So you don;t define all rational numbers..... – N. S. Feb 06 '16 at 01:02
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@N.S.: No, I don't miss the number $0$, as the first member of the pair is an integer and therefore can be $0$ (and also negative, BTW). – celtschk Jun 10 '16 at 17:20
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But even 'counting numbers' should include 0, no? I can count the colours of cars going past on a road and count 0 yellow ones and have a data point where the count of something being 0 makes sense. – JamEngulfer Oct 23 '16 at 20:56
@CharlesStewart I am taught the absolute contrary. – Mr Pie Jan 04 '18 at 05:43
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This answer doesn't mention any advantage/disadvantage of considering 0 natural number. Anyway, I was also told that 0 is natural in high school and university. The construction of natural numbers using set theory (that includes zero, which is the empty set) creates a very useful and fascinating symmetry that extends to algebra too. – ceztko Nov 20 '19 at 21:57
I occasionally run into this issue and always ask for clarification on someone's definition of "natural numbers". What's particularly annoying is that some people just religiously declare their definition as universally correct and then refuse to communicate clearly because "obviously you don't need to clarify that the natural numbers include/exclude 0". – Sjoerd Smit Oct 14 '22 at 09:10
@SjoerdSmit I agree, it should always be made explicit and clear in a given scenario. – Noldorin Oct 14 '22 at 14:12
@CharlesStewart curious, those definitions of “natural number” and “whole number” are exactly the reverse of what I (and several other commentators) recall learning in school, for me around age 9 or 10. (I remember it clearly because it was jarring to change my definition of “natural number” to the algebraic one when I started at university. So I used one definition for 8 years, and then the other one for the next 40, so far.) – Martin Kealey Apr 25 '25 at 00:51

Emanuele Paolini · Answer 2 · 2021-03-10T08:55:16.787

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I think that modern definitions include zero as a natural number. But sometimes, expecially in analysis courses, it could be more convenient to exclude it.

Pros of considering $0$ not to be a natural number:

generally speaking $0$ is not natural at all. It is special in so many respects;
people naturally start counting from $1$;
the harmonic sequence $1/n$ is defined for any natural number n;
the $1$st number is $1$;
in making limits, $0$ plays a role which is symmetric to $\infty$, and the latter is not a natural number.

Pros of considering $0$ a natural number:

the starting point for set theory is the emptyset, which can be used to represent $0$ in the construction of natural numbers; the number $n$ can be identified as the set of the first $n$ natural numbers;
computers start counting by $0$ (see the explanation of Dijkstra)
the rests in the integer division by a $n$ are $n$ different numbers starting from $0$ to $n-1$;
it is easier to exclude one defined element if we need naturals without zero; instead it is complicated to define a new element if we don't already have it;
integer, real and complex numbers include zero which seems much more important than $1$ in those sets (those sets are symmetric with respect to $0$);
there is a notion to define sets without $0$ (for example $\mathbb R_0$ or $\mathbb R_*$), or positive numbers ($\mathbb R_+$) but not a clear notion to define a set plus $0$;
the degree of a polynomial can be zero, as can be the order of a derivative;

I have seen children measure things with a ruler by aligning the $1$ mark instead of the $0$ mark. It is difficult to explain them why you have to start from $0$ when they are used to start counting from $1$. The marks in the rule identify the end of the centimeters, not the start, since the first centimeter goes from 0 to 1.

An example where counting from $1$ leads to somewhat wrong names is in the names of intervals between musical notes: the interval between C and F is called a fourth, because there are four notes: C, D, E, F. However the distance between C and F is actually three tones. This has the ugly consequence that a fifth above a fourth (4+3) is an octave (7) not a nineth! On the other hand if you put your first finger on the C note of a piano your fourth finger goes to the F note.

I would say that in the natural language the correspondence between cardinal numbers and ordinal numbers is off by one, thus distinguishing two sets of natural numbers, one starting from 0 and one starting from 1st. The 1st of January was day number $0$ of the new year. And zeroth has no meaning in the natural language...

edited Mar 10 '21 at 08:55

answered Jan 08 '15 at 10:21

Emanuele Paolini

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You do know that "natural" (in the pedestrian sense of the word, not the mathematical one) is completely subjective and dependent on your upbringing and social norms. It is unnatural to eat a cheeseburger in Israel (at least it was, say, 20 years ago) and it is unnatural to go out and drink beers during Passover. But do you consider cheeseburgers unnatural? In a few years, many children brought up in a vegan households will consider it unnatural to eat a cheeseburger as well (for different reasons). Others might find drinking to be unnatural. $0$ can be natural if you were taught it should be. – Asaf Karagila Jan 08 '15 at 10:30
@AsafKaragila Just saying, there has always been a huge non-Jewish population in Israel. There's a reason that three of the quarters of Jerusalem are the Muslim Quarter, the Christian Quarter, and the Armenian Quarter (also Christian). – Akiva Weinberger Jan 08 '15 at 11:21
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@columbus8myhw: I'm not trying to bring up a discussion about what is natural and what is not. I'm trying to point out that to say that "$0$ is not natural at all" depends on your upbringing and your social norms which may or may not consider it natural. But, hey, way to give into the stigma about mathematicians unable to transcend the minor mistakes in an analogy, and nitpick it apart! Kudos! – Asaf Karagila Jan 08 '15 at 11:24
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@AsafKaragila I agree with you. But I think (please confirm) that in every country a child is taught to attach numbers to things (i.e. counting) starting from 1. I think that also mathematicians do that: Problem #1 is the first problem in the list. In my experience only computer scientists do count from zero and only when talking with computers or other computer scientists... – Emanuele Paolini Jan 09 '15 at 12:02
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Emanuele, I'm not saying that in Israel, or somewhere else people start counting from $0$ as children. But my point is that what you might see as unnatural is only the result of your upbringing, so it is definitely not a valid mathematical or even a philosophical argument. The term "natural" loses its natural meaning when transferred to mathematics. There's nothing normal about $\Bbb R^{42}$ and there's nothing regular about $\omega_1$. Those are words, and if we insist on keeping their "natural language" meaning to them, then we're in trouble regarding most things in mathematics. – Asaf Karagila Jan 09 '15 at 12:05
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generally speaking 2 is unlike the other primes, it is special among primes in soo many different ways.... Should we exclude him fro the list of primes? – N. S. Feb 06 '16 at 01:05
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As a computer scientist, I think it's natural to count from zero. In fact, it's extremely strange if a programming language indexes arrays from 1, contrary to the norm. – mbomb007 Feb 16 '17 at 17:59
I agree with the children (but probably not for the same reason). On rulers where the 0 mark is at the edge of the ruler, I usually measure things by aligning the 1 mark because it can provide a more accurate reading. The end of the ruler can be worn and uneven, leading to measurement errors if I try to use it as a datum. I just have to remember to subtract 1 from the reading to get the actual measurement. (Sadly, I've been known to forget that last step on occasion.) – Ted Hopp Feb 25 '19 at 21:31
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computers start counting by 0 is not quite right; "computer scientists" generally start counting with 0, especially when indexing collections, but "computers" as such don't really count. – yoniLavi Mar 19 '19 at 17:47
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I appreciate the pros and cons list. It seems most useful to use N when including 0 and N* when excluding 0. The * is often used in sets and groups to denote the exclusion of 0. – Liam Bohl Jan 29 '21 at 23:16
Another pro of having zero: You can measure the cardinality of the empty set! – Atom Sep 03 '22 at 18:34
Just discovering this debate still exists, and I’m very surprised… Yes, absolutely, we start to count from 0, not from one if you consider that "counting" is to determine the cardinality of a set. « How many human missions on Neptune do you count? », « How many pterodactyl species are still living on earth? nowadays. The natural integer set has a lot less (to say the least) interesting properties if zero is not part of it. Just this one: The euclidean division is of practically no use without the zero. I’m astonished such a debate could still be a thing. – Stéphane Mar 17 '24 at 23:48

mau · Answer 3 · 2015-01-08T10:36:37.677

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There is no "official rule", it depends from what you want to do with natural numbers. Originally they started from $1$ because $0$ was not given the status of number.

Nowadays if you see $\mathbb{N}^+$ you may be assured we are talking about numbers from $1$ above; $\mathbb{N}$ is usually for numbers from $0$ above.

[EDIT: the original definitions of Peano axioms, as found in Arithmetices principia: nova methodo, may be found at https://archive.org/details/arithmeticespri00peangoog : look at it. ]

edited Jan 08 '15 at 10:36

answered Jul 21 '10 at 08:25

mau

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"$\mathbb{N}$ is usually for numbers from $0$ above." Can you point to evidence supporting this "usually"? – Jonas Meyer Aug 01 '13 at 04:01
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For evidence that the issue exists, http://mathworld.wolfram.com/NaturalNumber.html is a source; as for the "usually", I should dig my old books, I think. – mau Aug 01 '13 at 13:38
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That $\mathbb{N}$ usually includes zero. It is totally wrong. – Anixx Jan 08 '15 at 10:43
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Clapham-Nicholson, The Concise Oxford Dictionary of Mathematics (4th edition): «natural number - One of the numbers 1, 2, 3, ..., Some authors also include 0. The set of natural numbers is often denoted by N.» Eric Weisstein, Concise Encyclopedia of Mathematics (2nd ed) - «N: The SET of NATURAL NUMBERS (the POSITIVE INTEGERS Z : 1, 2, 3, ...; Sloane’s A000027), denoted N; also called the WHOLE NUMBERS . Like whole numbers, there is no general agreement on whether 0 should be included in the list of natural numbers.» – mau Jan 08 '15 at 11:34
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Interesting enough, Peano constructed naturals starting with 1, but construction of naturals with set theory starting with empty set 0 also satisfies Peano axioms. – ceztko Nov 20 '19 at 22:13
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@ceztko actually the two sets are isomorphic, it's just a matter or naming the numbers. – mau Nov 22 '19 at 10:25
I don't think it's only a matter of naming numbers. The numbers "mean" something which is made precise by cardinality. So if we have an ordered set of things say (a, b, c), it is naturally in bijection with (|{a}|, |{a, b}|, |{a, b, c}|) = (1, 2, 3) by taking cardinalities, i.e., counting. It is also in bijection with (0, 1, 2) but it is not "natural" as you have to subtract 1 from the cardinalities. – Pratyush Sarkar Aug 19 '21 at 19:24

score 31 · Answer 4 · answered Jun 22 '18 at 21:47

31

According to ISO 80000-2:2009: Quantities and Units - Part 2: Mathematical signs and symbols to be used in the natural sciences and technology, page 6;

$$\mathbb{N}=\{0,1,2,3,\ldots\}$$ $$\mathbb{N^*}=\{1,2,3,\ldots\}$$

answered Jun 22 '18 at 21:47

Evan Rosica

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i think Edsger W. Dijkstra has a useful comment on this. – robert bristow-johnson Oct 03 '18 at 18:51
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This should be the accepted answer. It is a convention, and an incredibly useful one in every area of mathematics (and computer science of course!) except for some parts of number theory where ${1,2,\dotsc}$ is the more fundamental object. – Martin Brandenburg Nov 24 '23 at 14:03
@MartinBrandenburg I have no real problem with $\mathbb N$ and $\mathbb N^$ for ${0,1,2,\ldots,}$ and ${1,2,\ldots}$ as in Evan Rosica's posted table, but for a large number of mathematicians, the symbol $R^$ when $R$ is a ring indicates the group of units, so $\mathbb Z^*$ is ${\pm1}$. Of course, for $R=\mathbb Q$ or $\mathbb R$ or $\mathbb C$ (or any field), this is equivalent to the set of non-zero elements. But for the integers, I'd certainly prefer $\mathbb Z_{\ne0}$ for the set of non-zero elements. – Joe Silverman Mar 15 '24 at 20:00

score 20 · Answer 5 · answered Feb 05 '16 at 16:59

20

These lecture notes from a combinatorics course given for many years by N.G. de Bruijn suggest a helpful alternative:

Due to the confusion caused by N. Bourbaki about the natural numbers, we feel obliged to define: $$\begin{align}\Bbb N_0 & = \{0,1,2,\ldots\}\quad \text{ and } \\ \Bbb N_1 & = \{1,2,3,\ldots\}. \end{align}$$

(Page 4)

answered Feb 05 '16 at 16:59

MJD

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Yeah. I had a professor who used $\mathbb{N}{k} = { k, k + 1, k + 2, \ldots }$ to talk about counting integers from $k$ on (he often used $\mathbb{N}{2}$ to talk about base-$b$ expansions), a practice I adopted (though you'll wanna write somewhere in there what you mean). – AJY Feb 06 '16 at 00:56
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@AJY, its worth pointing out that if we define $\mathbb{N} = {0,1,2,\ldots}$, then: $$\mathbb{N}_k = k+\mathbb{N}.$$ – goblin GONE Apr 03 '16 at 05:06
@goblin Correct, at least when we suppose $0 \in \mathbb{N}$. – AJY Apr 03 '16 at 05:07
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The dig at N. Bourbaki is very implicit and curious, as if there were inconsistency in their books as to whether $0\in\mathbf{N}$ (which would be conceivable, given that they are written by many different people). As far as I can tell, Bourbaki unanimously has $0\in\mathbf{N}$. There are just some differences in the ways in winch they (don't explicitly) say this, compare for instance Théorie des Ensembles Ch III, §4 1 ..."un cardinal fini s'appelle aussi entier naturel" with Algèbre Ch. I §2 5 I "Considérons le monoïde commutatif N des entiers naturels". – Marc van Leeuwen Mar 02 '23 at 13:22
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... The only inconsistency I can find is that the (rather confusing) numbering of their books, chapters, sections, paragraphs never starts with $0$, but this can at least in part be defended by the fact that the numbering often uses Roman numerals. – Marc van Leeuwen Mar 02 '23 at 13:27
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@MarcvanLeeuwen It implies that until Bourbaki, everyone, including Peano, started $\mathbb N$ with $1$. Bourbaki changed the definition. – SRobertJames Dec 04 '24 at 19:40

score 15 · Answer 6 · edited Mar 15 '13 at 19:04

15

There are the two definitions, as you say. However the set of strictly positive numbers being the natural numbers is actually the older definition. Inclusion of $0$ in the natural numbers is a definition for them that first occurred in the 19th century.

The Peano Axioms for natural numbers take $0$ to be one though, so if you are working with these axioms (and a lot of natural number theory does) then you take $0$ to be a natural number.

edited Mar 15 '13 at 19:04

Dominic Michaelis

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answered Jul 21 '10 at 08:03

workmad3

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Bourbaki included zero in 1935. That's not all that recent... – Jul 21 '10 at 10:08
More recently, rather than actually recent... but point taken, I'll revise the wording :) – workmad3 Jul 21 '10 at 11:57
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"take $0$ to be one" is confusing phrasing. – Jonas Meyer Aug 01 '13 at 04:00
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@JonasMeyer: Actually it makes sense in a non-trivial way: From the axioms, the natural numbers are basically a set with a first number, and then a successor for each number. At that point, you cannot actually distinguish between the natural numbers starting at $0$ and the natural numbers starting at $1$, except for the (completely arbitrary) name for the initial number. Basically, it is the different definition for addition and multiplication which distinguishes the two choices. – celtschk Aug 01 '13 at 16:57
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Now if you say "one" is the name of the initial natural number, then "take $0$ to be one" would be interpreted as "take $0$ to be the initial number", that is, "start the natural numbers with $0$". – celtschk Aug 01 '13 at 17:00
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@celtschk: Interesting point, thank you. It makes sense also in the way it was probably intended, namely "$0$ is one of the natural numbers," i.e., just as $15$ is one. In any case it is not a real problem, just potentially confusing. – Jonas Meyer Aug 01 '13 at 20:30
"The Peano Axioms for natural numbers take 0 to be (a natural number)" is a bit misleading. I have just checked Arithmetices principia, nova methodo exposita (G. Peano, 1889) and it does not do this. His Axiom 1 is "$1\epsilon N$". He doesn't have a special notation for successor, using "$a+1$" to mean the successor of $a$, and there is nothing like $a+0=a$. The definition of addition is that $a+(b+1) = (a+b)+1$. In more modern notation we would write $a+\operatorname{Suc}(b) = \operatorname{Suc}(a+b)$. Note that this doesn't explicitly cover the case of adding $a$ to $1$. – MJD Dec 04 '24 at 20:07

score 6 · Answer 7 · answered Dec 30 '21 at 00:44

The Peano-Dedekind axioms (as used in proving propositions by use of the Principle of Mathematical Induction) define the $\mathbb{N}$ as either $\mathbb{N}$ = $\mathbb{Z^+} \cup \text{0} = \text{{0, 1, 2, ...}}$ or $\mathbb{N} = \mathbb{Z^+} = \text{{1, 2, 3, ...}}$, that is, it depends on the context (usually this "context" may be seen from the given proposition to be proved, at least in the case of using PMI).

score 5 · Answer 8 · edited Mar 15 '13 at 19:08

5

I remember all of my courses at University using only positive integers (not including $0$) for the Natural Numbers. It's possible that they had come to an agreement amongst the Maths Faculty, but during at least two courses we generated the set of natural numbers in ways that wouldn't make sense if $0$ was included.

One involved the cardinality of Sets of Sets, the other defined the natural numbers in terms of the number $1$ and addition only ($0$ and Negative Integers come into the picture later when you define an inverse to addition).

As a result when teaching the difference between Integers and Natural Numbers I always define $0$ as an integer that isn't a Natural Number.

edited Mar 15 '13 at 19:08

Dominic Michaelis

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answered Jul 21 '10 at 09:52

Amos

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Obviously, defining ℕ from 0 and addition also works perfectly. I don't know what difference would 0 make to calculating the cardinality of P(ℕ) either. – badp Jul 21 '10 at 12:08
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The cardinality of sets of sets can certainly be $0$: All members of the empty set are sets. Indeed, in ZF all sets are sets of sets. – celtschk Aug 01 '13 at 17:17
@badp Ok, you have 0 and addition. 0+0=0. Where's the rest of N? – Evan Rosica Oct 17 '16 at 02:54
Defining the natural numbers as the closure of (the set containing) $1$ under arbitrary sums includes $0$, because $0$ is the empty sum of $1$s. – LSpice Jan 30 '24 at 00:36

MyMathYourMath · Answer 9 · 2025-11-09T02:05:41.217

Excellent question and one which I have wondered about for years since ambiguity shouldn’t exist in mathematics. In short, I just denote these as $\Bbb{Z}^+,\Bbb{Z}^{\geq}$.

However, it really depends on context. For instance in set theory,

$$\Bbb{N}=\{0,1,2,…\}.$$

This is so that you can inductively or recursively define all the naturals and you include the empty set. If you’re studying analysis, typically

$$\Bbb{N}=\{1,2,3,…\}.$$

Or equivalently one could write $\Bbb{Z}^+$ for $1,2,…$ since $0$ is neither positive nor negative.

I myself am an educator and in the field when dealing with grade school children their textbooks define the naturals as starting with $1$ and the whole numbers, you guessed it, denoted $\Bbb{W}$, as the set of naturals starting at zero. Where they always remind them zero is a whole number because it has a hole in it. This is a crucial point to bring up as many of us have children who go crazy when the math isn’t mathing.

To avoid confusion I say we put

$$\Bbb{Z}^+=\{1,2,3,...\}$$

For starting at $1$. And

$$\Bbb{Z}^{\geq 0}=\{0,1,2,...\},$$

For starting at zero and call it a day. You have to admit, calling them these two avoids all ambiguity. Or so I like to say “zee plus and zee geque”. Sorry bit of poor math humor there.

I’ll add one thing, depending on your starting place, the answer to infinite sums can be different. And for some sequences you won’t be allowed to plug in say zero so it will begin at $1$. That is, $$\sum_{n=0}^\infty\frac{1}{2^n}=2,$$ but $$\sum_{n=1}^\infty\frac{1}{2^n}=1.$$ So if you're finding a sum it really matters but if you're doing measure theory and using an $\frac{\epsilon}{2^n}$ argument it won't matter because $\epsilon>0$ is arbitrary.

score -2 · Answer 10 · answered Aug 04 '23 at 14:21

This may also depend on the country. I do not know about the prevalent convention in the United States, but generally speaking:

French authors do include $0$ in the natural numbers,
German authors do not include $0$ in the natural numbers.

I am not aware of any comprehensive survey on the topic, even though it tends to be contested.

score -4 · Answer 11 · answered Oct 10 '21 at 03:44

Peano used 1 as the first natural number in his arithmetic theory. Geometric: If a line is breadthless with infinite points, then the first line you can create is a 1. Written 1 if it were a map you could scale it. This 1 can then be used as a metric to index a length of each successive natural number. On a Cartesian Graph 0 is used to define one end point of the 1, but without the second end point at 1 there is no length, and no number according to some. Also consider, if you were creating a set of lines each representing a natural number, what would 0 look like? There would be no length to 0, and hence it is not a member of the set of lines.

Is $0$ a natural number?

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