# Definition:Natural Numbers

## Informal Definition

The natural numbers are the counting numbers.

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

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

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

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

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

## Axiomatic Definition

### Natural Numbers form Naturally Ordered Semigroup

The natural numbers under addition form an algebraic structure $\left({\N, +, \le}\right)$ which is a naturally ordered semigroup.

### Natural Numbers as 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:

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

### Peano's Axioms Uniquely Define Natural Numbers

Peano's Axioms uniquely define the set of natural numbers.

That is:

• not only do the natural numbers satisfy Peano's Axioms;
• but conversely, any set that satisfies Peano's Axioms also satisfies all the properties held by the set $\N$ of Natural Numbers.

Thus the structure of the set $\N$ of natural numbers is characterised completely by these axioms:

 $$(P1):$$ $$\displaystyle \exists 0:$$ $$\displaystyle 0 \in \N$$ $$(P2):$$ $$\displaystyle \forall n \in \N:$$ $$\displaystyle \exists n' \in \N$$ $$(P3):$$ $$\displaystyle \neg \left({\exists n \in \N: n' = 0}\right)$$ $$(P4):$$ $$\displaystyle \forall m, n \in \N:$$ $$\displaystyle n' = m' \implies n = m$$ $$(P5):$$ $$\displaystyle \forall A \subseteq \N:$$ $$\displaystyle \left({0 \in A \land \left({n \in A \implies n' \in A}\right)} \right) \implies A = \N$$

These can be expressed in natural language as:

 $$(P1):$$ There exists a natural number $0$. $$(P2):$$ For every natural number $n$ there exists another, known as the successor of $n$. $$(P3):$$ No number has $0$ as its successor. $$(P4):$$ If two numbers have the same successor, they are the same number. Or: different numbers have different successors. $$(P5):$$ A subset of the natural numbers with $0$ in it, such that it has the successor of every number in it, is the same set as the natural numbers.

In this context, the element $n'$ denotes the (immediate) successor element of $n$, which (in the context of the natural numbers) is understood as meaning $n + 1$.

### Axiom Schema for 1-Based Natural Numbers

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

### Natural Numbers as Cardinals

The natural numbers $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$ can be defined as the set of cardinals.

### 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

In the field of computer science, a natural number is usually referred to as an unsigned number, which arises from the fact that it has no positive or negative sign attached.

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.

Paul R. Halmos: Naive Set Theory (1960) uses $\omega$.