Definition:Natural Numbers

This page is about everyday numbers used for counting. For the representation of $\N$ as used in set theory, see Definition:Finite Ordinal.

Informal Definition

The natural numbers are the counting numbers.

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

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

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

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

The set of natural numbers is one of the most important sets in mathematics.

Axiomatization

Peano's Axioms

Peano's Axioms are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: s \left({n}\right) = n + 1$ 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. The other three are as follows:

 $(P3)$ $:$ $\displaystyle \forall m, n \in P:$ $\displaystyle \map s m = \map s n \implies m = n$ $s$ is injective $(P4)$ $:$ $\displaystyle \forall n \in P:$ $\displaystyle \map s n \ne 0$ $0$ is not in the image of $s$ $(P5)$ $:$ $\displaystyle \forall A \subseteq P:$ $\displaystyle \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P$ 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 $\left({S, \circ, \preceq}\right)$ satisfying:

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

1-Based Natural Numbers

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

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

Construction

Elements of Minimal Infinite Successor Set

Let $\omega$ denote the minimal infinite successor 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:

$0 := \O = \set {}$
$1 := 0^+ = 0 \cup \set 0 = \set 0$
$2 := 1^+ = 1 \cup \set 1 = \set {0, 1}$
$3 := 2^+ = 2 \cup \set 2 = \set {0, 1, 2}$
$\vdots$

Natural Numbers in Real Numbers

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

Let $\mathcal I$ be the set of all inductive sets in $\R$.

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

$\N := \displaystyle \bigcap \mathcal I$

where $\displaystyle \bigcap$ 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 1951: Nathan Jacobson: Lectures in Abstract Algebra: I. Basic Concepts 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.

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

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 see

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

Historical Note

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.