Definition:Natural Numbers

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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 $\N$:

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

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


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

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

This sequence is A000027 in the On-Line Encyclopedia of Integer Sequences (N. J. A. Sloane (Ed.), 2008).


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 $s: \N \to \N: s(n) = n + 1$ 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. The other three are as follows:

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

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


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 $\varnothing$ and successor sets, we thus define:

$0 := \varnothing = \left\{{}\right\}$
$1 := 0^+ = 0 \cup \left\{{0}\right\} = \left\{{0}\right\}$
$2 := 1^+ = 1 \cup \left\{{1}\right\} = \left\{{0, 1}\right\}$
$3 := 2^+ = 2 \cup \left\{{2}\right\} = \left\{{0, 1, 2}\right\}$
$\vdots$


Natural Numbers in Real Numbers

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

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


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

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

where $\displaystyle \bigcap$ denotes intersection.


Also known as

First, note that some sources use a different style of letter from $\N$: you will find $N$, $\mathbf N$, etc. However, $\N$ is becoming more commonplace and universal nowadays.


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


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

Either is valid, and as long as it is clear which is which, it does not matter which is used. However, using $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ 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 $\left\{{1, 2, 3, \ldots}\right\}$ sometimes refer to $\left\{{0, 1, 2, 3, \ldots}\right\}$ as the positive (or non-negative) integers $\Z_{\ge 0}$.


The following notations are sometimes used:

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


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

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

under which notational convention $\N_0 = \varnothing$ and $\N_1 = \left\{{0}\right\}$.

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: I. Basic Concepts (1951) uses $P = \left\{{1, 2, 3, \ldots}\right\}$.

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 $\omega$, Paul R. Halmos: Naive Set Theory (1960) 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, $\left({\N, \le}\right)$ where $\le$ is the usual ordering on the natural numbers.


Also see


Sources