# 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 $\N$:

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

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

- Definition:Natural Number Addition
- Results about
**natural numbers as an abstract algebraical concept**can be found here.

## Sources

- Nathan Jacobson:
*Lectures in Abstract Algebra: I. Basic Concepts*(1951)... (previous)... (next): Introduction $\S 4$: The natural numbers - Paul R. Halmos:
*Naive Set Theory*(1960)... (previous)... (next): $\S 11$: Numbers - W.E. Deskins:
*Abstract Algebra*(1964)... (previous)... (next): $\S 2.1$ - Murray R. Spiegel:
*Theory and Problems of Complex Variables*(1964)... (next): $1$: Complex Numbers: The Real Number System: $1$ - Seth Warner:
*Modern Algebra*(1965)... (previous)... (next): $\S 1$ - Allan Clark:
*Elements of Abstract Algebra*(1971)... (previous)... (next): $\S 15$ - Robert H. Kasriel:
*Undergraduate Topology*(1971)... (previous)... (next): $\S 1.8$: Collections of Sets: Definition $8.4$ - A.G. Howson:
*A Handbook of Terms used in Algebra and Analysis*(1972)... (previous)... (next): $\S 4$: Number systems $\text{I}$ - T.S. Blyth:
*Set Theory and Abstract Algebra*(1975)... (previous)... (next): $\S 1$ - W.A. Sutherland:
*Introduction to Metric and Topological Spaces*(1975)... (previous)... (next): Notation and Terminology - K.G. Binmore:
*Mathematical Analysis: A Straightforward Approach*(1977)... (previous)... (next): $\S 1.2$ - Thomas A. Whitelaw:
*An Introduction to Abstract Algebra*(1978)... (previous)... (next): $\S 2 \ \text{(b)}$ - P.M. Cohn:
*Algebra Volume 1*(2nd ed., 1982)... (previous)... (next): $\S 2.1$: The integers - H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*(1996)... (previous)... (next): Appendix $\text{A}.1$: Sets - Michael R.A. Huth and Mark D. Ryan:
*Logic in Computer Science: Modelling and reasoning about systems*(2000)... (previous)... (next): $\S 1.2.5$: An aside: proof by contradiction