Definition:Natural Numbers

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This page is about everyday numbers used for counting. For other uses, see Finite Ordinal.

Informal Definition

The natural numbers are the numbers used for counting.


The set of natural numbers is denoted $\N$:

$\N = \set {0, 1, 2, 3, \ldots}$

This sequence is A001477 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


Non-Zero Natural Numbers

The natural numbers are often defined so as to exclude zero:

The set $\N \setminus \set 0$ is denoted $\N_{>0}$:

$\N_{>0} = \set {1, 2, 3, \ldots}$


Axiomatization

Peano's Axioms

Peano's axioms are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ and $0$ as an element of $\N$.


Let there be given a set $P$, a mapping $s: P \to P$, and a distinguished element $0$.

Historically, the existence of $s$ and the existence of $0$ were considered the first two of Peano's axioms:

\((\text P 1)\)   $:$   \(\ds 0 \in P \)    $0$ is an element of $P$      
\((\text P 2)\)   $:$     \(\ds \forall n \in P:\) \(\ds \map s n \in P \)    For all $n \in P$, its successor $\map s n$ is also in $P$      

The other three are as follows:

\((\text P 3)\)   $:$     \(\ds \forall m, n \in P:\) \(\ds \map s m = \map s n \implies m = n \)    $s$ is injective      
\((\text P 4)\)   $:$     \(\ds \forall n \in P:\) \(\ds \map s n \ne 0 \)    $0$ is not in the image of $s$      
\((\text P 5)\)   $:$     \(\ds \forall A \subseteq P:\) \(\ds \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \)    Principle of Mathematical Induction:      
Any subset $A$ of $P$, containing $0$ and      
closed under $s$, is equal to $P$      


Naturally Ordered Semigroup

The concept of a naturally ordered semigroup is intended to capture the behaviour of the natural numbers $\N$, addition $+$ and the ordering $\le$ as they pertain to $\N$.


Naturally Ordered Semigroup Axioms

A naturally ordered semigroup is a (totally) ordered commutative semigroup $\struct {S, \circ, \preceq}$ satisfying:

\((\text {NO} 1)\)   $:$   $S$ is well-ordered by $\preceq$      \(\ds \forall T \subseteq S:\) \(\ds T = \O \lor \exists m \in T: \forall n \in T: m \preceq n \)      
\((\text {NO} 2)\)   $:$   $\circ$ is cancellable in $S$      \(\ds \forall m, n, p \in S:\) \(\ds m \circ p = n \circ p \implies m = n \)      
\(\ds p \circ m = p \circ n \implies m = n \)      
\((\text {NO} 3)\)   $:$   Existence of product      \(\ds \forall m, n \in S:\) \(\ds m \preceq n \implies \exists p \in S: m \circ p = n \)      
\((\text {NO} 4)\)   $:$   $S$ has at least two distinct elements      \(\ds \exists m, n \in S:\) \(\ds m \ne n \)      


1-Based Natural Numbers

The following axioms capture the behaviour of $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations $+$ and $\times$ as they pertain to $\N_{>0}$:

\((\text A)\)   $:$     \(\ds \exists_1 1 \in \N_{> 0}:\) \(\ds a \times 1 = a = 1 \times a \)      
\((\text B)\)   $:$     \(\ds \forall a, b \in \N_{> 0}:\) \(\ds a \times \paren {b + 1} = \paren {a \times b} + a \)      
\((\text C)\)   $:$     \(\ds \forall a, b \in \N_{> 0}:\) \(\ds a + \paren {b + 1} = \paren {a + b} + 1 \)      
\((\text D)\)   $:$     \(\ds \forall a \in \N_{> 0}, a \ne 1:\) \(\ds \exists_1 b \in \N_{> 0}: a = b + 1 \)      
\((\text E)\)   $:$     \(\ds \forall a, b \in \N_{> 0}:\) \(\ds \)Exactly one of these three holds:\( \)      
\(\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \)      
\((\text F)\)   $:$     \(\ds \forall A \subseteq \N_{> 0}:\) \(\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \)      


Construction

Von Neumann Construction of Natural Numbers

Let $\omega$ denote the minimally inductive set.

The natural numbers can be defined as the elements of $\omega$.

Following Definition 2 of $\omega$, this amounts to defining the natural numbers as the finite ordinals.


In terms of the empty set $\O$ and successor sets, we thus define:

\(\ds 0\) \(:=\) \(\ds \O = \set {}\)
\(\ds 1\) \(:=\) \(\ds 0^+ = 0 \cup \set 0 = \set 0\)
\(\ds 2\) \(:=\) \(\ds 1^+ = 1 \cup \set 1 = \set {0, 1}\)
\(\ds 3\) \(:=\) \(\ds 2^+ = 2 \cup \set 2 = \set {0, 1, 2}\)
\(\ds \) \(\vdots\) \(\ds \)
\(\ds n + 1\) \(:=\) \(\ds n^+ = n \cup \set n\)


Inductive Set Definition for Natural Numbers

Let $x$ be a set which is an element of every inductive set.

Then $x$ is a natural number.


Inductive Set Definition for Natural Numbers in Real Numbers

Let $\R$ be the set of real numbers.

Let $\II$ be the set of all inductive sets defined as subsets of $\R$.


Then the natural numbers $\N$ are defined as:

$\N := \ds \bigcap \II$

where $\ds \bigcap$ denotes intersection.


Notation

Some sources use a different style of letter from $\N$: you will find $N$, $\mathbf N$, etc. However, $\N$ is commonplace and all but universal.


The usual symbol for denoting $\set {1, 2, 3, \ldots}$ is $\N^*$, but the more explicit $\N_{>0}$ is standard on $\mathsf{Pr} \infty \mathsf{fWiki}$.


Some authors refer to $\set {0, 1, 2, 3, \ldots}$ as $\tilde \N$, and refer to $\set {1, 2, 3, \ldots}$ as $\N$.

Either is valid, and as long as it is clear which is which, it does not matter which is used. However, using $\N = \set {0, 1, 2, 3, \ldots}$ is a more modern approach, particularly in the field of computer science, where starting the count at zero is usual.

Treatments which consider the natural numbers as $\set {1, 2, 3, \ldots}$ sometimes refer to $\set {0, 1, 2, 3, \ldots}$ as the positive (or non-negative) integers $\Z_{\ge 0}$.


The following notations are sometimes used:

$\N_0 = \set {0, 1, 2, 3, \ldots}$
$\N_1 = \set {1, 2, 3, \ldots}$


However, beware of confusing this notation with the use of $\N_n$ as the initial segment of the natural numbers:

$\N_n = \set {0, 1, 2, \ldots, n - 1}$

under which notational convention $\N_0 = \O$ and $\N_1 = \set 0$.

So it is important to ensure that it is understood exactly which convention is being used.


The use of $\N$ or its variants is not universal, either.

Some sources, for example Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ($1951$) uses $P = \set {1, 2, 3, \ldots}$.

This may stem from the fact that Jacobson's presentation starts with Peano's axioms.

On the other hand, it may just be because $P$ is the first letter of positive.

Similarly, Undergraduate Topology by Robert H. Kasriel uses $\mathbf P$, which he describes as the set of all positive integers.


Based on defining $\N$ as being the minimally inductive set $\omega$, 1960: Paul R. Halmos: Naive Set Theory suggests using $\omega$ for the set of natural numbers.

This convention is followed by Raymond M. Smullyan and Melvin Fitting in their Set Theory and the Continuum Problem.

This use of $\omega$ is usually seen for the order type of the natural numbers, that is, $\struct {\N, \le}$ where $\le$ is the usual ordering on the natural numbers.


Also known as

The natural numbers, particularly the set $\set {1, 2, 3, \ldots}$ of non-zero natural numbers, are also known as counting numbers, especially in elementary school.

Some sources refer to them as positive integers, often with the implicit understanding that this means strictly positive integers, without actually being specific.

Some sources refer to the natural numbers as whole numbers, but that term is also often given to the integers.

The term proper numbers can sometimes be seen in popular literature.


Also see

  • Results about natural numbers as an abstract algebraical concept can be found here.


Historical Note

The natural numbers were the first numbers to be considered.

Their earliest use was in the sense of ordinal numbers, when they were used for counting.

The origin of the name natural numbers (considered by some authors to be a misnomer) originates with the Ancient Greeks, for whom the only numbers were the strictly positive integers $1, 2, 3, \ldots$


It is customary at this stage to quote the famous epigram of Leopold Kronecker, translated from the German in various styles, for example:

God created the natural numbers, and all the rest is the work of man.

The exact word he used was Zahlen, which some translate as integers; the distinction is of little importance in this context.


The intuitionist viewpoint has that the natural numbers can be accepted as a primitive concept, despite the fact that they are infinite in number.


Linguistic Note

The words for the individual natural numbers in ancient languages which have now been supplanted by newer ones have in some cases survived in remote places for special purposes.

The traditional system of numbers used for counting sheep in certain locales in the British Isles is one example:

\((1)\)   $:$   wan      
\((2)\)   $:$   twan      
\((3)\)   $:$   tethera      
\((4)\)   $:$   methera      
\((5)\)   $:$   pimp      
\((6)\)   $:$   sethera      
\((7)\)   $:$   lethera      
\((8)\)   $:$   hovera      
\((9)\)   $:$   dovera      
\((10)\)   $:$   dick      
\((11)\)   $:$   wanadick      
\((12)\)   $:$   twanadick      
\((13)\)   $:$   tetheradick      
\((14)\)   $:$   metheradick      
\((15)\)   $:$   pimpdick      
\((16)\)   $:$   setheradick      
\((17)\)   $:$   letheradick      
\((18)\)   $:$   hoveradick      
\((19)\)   $:$   doveradick      
\((20)\)   $:$   bumfit      
\((21)\)   $:$   wanabumfit      

and so on.


Sources

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