# 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. *

## Contents

## Informal Definition

The **natural numbers** are the counting numbers.

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).

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

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

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 $s: \N \to \N: s \left({n}\right) = n + 1$ 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.*

## Sources

- 1951: Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*... (previous) ... (next): Introduction $\S 4$: The natural numbers - 1960: Walter Ledermann:
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*Elements of Abstract Algebra*... (previous) ... (next): Chapter $1$: Mappings: $\S 15$ - 1971: Robert H. Kasriel:
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*Taming the Infinite*... (previous) ... (next): Chapter $3$: Notations and Numbers