Definition:Natural Numbers
This page is about everyday numbers used for counting. For other uses, see Finite Ordinal.
Informal Definition
The natural numbers are the numbers used for counting.
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).
Non-Zero Natural Numbers
The natural numbers are often defined so as to exclude zero:
The set $\N \setminus \set 0$ is denoted $\N_{>0}$:
- $\N_{>0} = \set {1, 2, 3, \ldots}$
Axiomatization
Peano's Axioms
Peano's axioms are intended to reflect the intuition behind $\N$, the mapping $s: \N \to \N: \map s n = n + 1$ 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:
\((\text P 1)\) | $:$ | \(\ds 0 \in P \) | $0$ is an element of $P$ | ||||||
\((\text P 2)\) | $:$ | \(\ds \forall n \in P:\) | \(\ds \map s n \in P \) | For all $n \in P$, its successor $\map s n$ is also in $P$ |
The other three are as follows:
\((\text P 3)\) | $:$ | \(\ds \forall m, n \in P:\) | \(\ds \map s m = \map s n \implies m = n \) | $s$ is injective | |||||
\((\text P 4)\) | $:$ | \(\ds \forall n \in P:\) | \(\ds \map s n \ne 0 \) | $0$ is not in the image of $s$ | |||||
\((\text P 5)\) | $:$ | \(\ds \forall A \subseteq P:\) | \(\ds \paren {0 \in A \land \paren {\forall z \in A: \map s z \in A} } \implies A = P \) | 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 $\struct {S, \circ, \preceq}$ satisfying:
\((\text {NO} 1)\) | $:$ | $S$ is well-ordered by $\preceq$ | \(\ds \forall T \subseteq S:\) | \(\ds T = \O \lor \exists m \in T: \forall n \in T: m \preceq n \) | |||||
\((\text {NO} 2)\) | $:$ | $\circ$ is cancellable in $S$ | \(\ds \forall m, n, p \in S:\) | \(\ds m \circ p = n \circ p \implies m = n \) | |||||
\(\ds p \circ m = p \circ n \implies m = n \) | |||||||||
\((\text {NO} 3)\) | $:$ | Existence of product | \(\ds \forall m, n \in S:\) | \(\ds m \preceq n \implies \exists p \in S: m \circ p = n \) | |||||
\((\text {NO} 4)\) | $:$ | $S$ has at least two distinct elements | \(\ds \exists m, n \in S:\) | \(\ds m \ne n \) |
1-Based Natural Numbers
The following axioms capture the behaviour of $\N_{>0}$, the element $1 \in \N_{>0}$, and the operations $+$ and $\times$ as they pertain to $\N_{>0}$:
\((\text A)\) | $:$ | \(\ds \exists_1 1 \in \N_{> 0}:\) | \(\ds a \times 1 = a = 1 \times a \) | ||||||
\((\text B)\) | $:$ | \(\ds \forall a, b \in \N_{> 0}:\) | \(\ds a \times \paren {b + 1} = \paren {a \times b} + a \) | ||||||
\((\text C)\) | $:$ | \(\ds \forall a, b \in \N_{> 0}:\) | \(\ds a + \paren {b + 1} = \paren {a + b} + 1 \) | ||||||
\((\text D)\) | $:$ | \(\ds \forall a \in \N_{> 0}, a \ne 1:\) | \(\ds \exists_1 b \in \N_{> 0}: a = b + 1 \) | ||||||
\((\text E)\) | $:$ | \(\ds \forall a, b \in \N_{> 0}:\) | \(\ds \)Exactly one of these three holds:\( \) | ||||||
\(\ds a = b \lor \paren {\exists x \in \N_{> 0}: a + x = b} \lor \paren {\exists y \in \N_{> 0}: a = b + y} \) | |||||||||
\((\text F)\) | $:$ | \(\ds \forall A \subseteq \N_{> 0}:\) | \(\ds \paren {1 \in A \land \paren {z \in A \implies z + 1 \in A} } \implies A = \N_{> 0} \) |
Construction
Von Neumann Construction of Natural Numbers
Let $\omega$ denote the minimally inductive 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:
\(\ds 0\) | \(:=\) | \(\ds \O = \set {}\) | ||||||||||||
\(\ds 1\) | \(:=\) | \(\ds 0^+ = 0 \cup \set 0 = \set 0\) | ||||||||||||
\(\ds 2\) | \(:=\) | \(\ds 1^+ = 1 \cup \set 1 = \set {0, 1}\) | ||||||||||||
\(\ds 3\) | \(:=\) | \(\ds 2^+ = 2 \cup \set 2 = \set {0, 1, 2}\) | ||||||||||||
\(\ds \) | \(\vdots\) | \(\ds \) | ||||||||||||
\(\ds n + 1\) | \(:=\) | \(\ds n^+ = n \cup \set n\) |
Inductive Set Definition for Natural Numbers
Let $x$ be a set which is an element of every inductive set.
Then $x$ is a natural number.
Inductive Set Definition for Natural Numbers in Real Numbers
Let $\R$ be the set of real numbers.
Let $\II$ be the set of all inductive sets defined as subsets of $\R$.
Then the natural numbers $\N$ are defined as:
- $\N := \ds \bigcap \II$
where $\ds \bigcap$ 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 Nathan Jacobson: Lectures in Abstract Algebra: Volume $\text { I }$: Basic Concepts ($1951$) 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.
Similarly, Undergraduate Topology by Robert H. Kasriel uses $\mathbf P$, which he describes as the set of all positive integers.
Based on defining $\N$ as being the minimally inductive set $\omega$, 1960: Paul R. Halmos: Naive Set Theory suggests using $\omega$ for the set of natural numbers.
This convention is followed by Raymond M. Smullyan and Melvin Fitting in their Set Theory and the Continuum Problem.
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 known as
The natural numbers, particularly the set $\set {1, 2, 3, \ldots}$ of non-zero natural numbers, are also known as counting numbers, especially in elementary school.
Some sources refer to them as positive integers, often with the implicit understanding that this means strictly positive integers, without actually being specific.
Some sources refer to the natural numbers as whole numbers, but that term is also often given to the integers.
The term proper numbers can sometimes be seen in popular literature.
Also see
- Results about natural numbers as an abstract algebraical concept can be found here.
Historical Note
The natural numbers were the first numbers to be considered.
Their earliest use was in the sense of ordinal numbers, when they were used for counting.
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.
The exact word he used was Zahlen, which some translate as integers; the distinction is of little importance in this context.
The intuitionist viewpoint has that the natural numbers can be accepted as a primitive concept, despite the fact that they are infinite in number.
Linguistic Note
The words for the individual natural numbers in ancient languages which have now been supplanted by newer ones have in some cases survived in remote places for special purposes.
The traditional system of numbers used for counting sheep in certain locales in the British Isles is one example:
\((1)\) | $:$ | wan | |||||||
\((2)\) | $:$ | twan | |||||||
\((3)\) | $:$ | tethera | |||||||
\((4)\) | $:$ | methera | |||||||
\((5)\) | $:$ | pimp | |||||||
\((6)\) | $:$ | sethera | |||||||
\((7)\) | $:$ | lethera | |||||||
\((8)\) | $:$ | hovera | |||||||
\((9)\) | $:$ | dovera | |||||||
\((10)\) | $:$ | dick | |||||||
\((11)\) | $:$ | wanadick | |||||||
\((12)\) | $:$ | twanadick | |||||||
\((13)\) | $:$ | tetheradick | |||||||
\((14)\) | $:$ | metheradick | |||||||
\((15)\) | $:$ | pimpdick | |||||||
\((16)\) | $:$ | setheradick | |||||||
\((17)\) | $:$ | letheradick | |||||||
\((18)\) | $:$ | hoveradick | |||||||
\((19)\) | $:$ | doveradick | |||||||
\((20)\) | $:$ | bumfit | |||||||
\((21)\) | $:$ | wanabumfit |
and so on.
Sources
- 1964: J. Hunter: Number Theory ... (previous) ... (next): Chapter $\text {I}$: Number Systems and Algebraic Structures: $1$. Introduction
- 1965: Seth Warner: Modern Algebra ... (previous) ... (next): Chapter $\text I$: Algebraic Structures: $\S 1$: The Language of Set Theory
- 1972: A.G. Howson: A Handbook of Terms used in Algebra and Analysis ... (previous) ... (next): $\S 4$: Number systems $\text{I}$
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 1$. Sets; inclusion; intersection; union; complementation; number systems
- 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): Notation and Terminology
- 1978: Thomas A. Whitelaw: An Introduction to Abstract Algebra ... (previous) ... (next): $\S 2$: Introductory remarks on sets: $\text{(b)}$
- 1982: Alan G. Hamilton: Numbers, Sets and Axioms ... (next): $\S 1$: Numbers: $1.1$ Natural Numbers and Integers
- 1994: H.E. Rose: A Course in Number Theory (2nd ed.) ... (previous) ... (next): $1$ Divisibility: $1.1$ The Euclidean algorithm and unique factorization
- 1996: H. Jerome Keisler and Joel Robbin: Mathematical Logic and Computability ... (previous) ... (next): Appendix $\text A$: Sets and Functions: $\text{A}.1$: Sets
- 1999: András Hajnal and Peter Hamburger: Set Theory ... (previous) ... (next): $1$. Notation, Conventions: $2$
- 2000: Michael R.A. Huth and Mark D. Ryan: Logic in Computer Science: Modelling and reasoning about systems ... (previous) ... (next): $\S 1.2.5$: An aside: proof by contradiction
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 1$ What is infinity?
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $1$: General Background: $\S 5$ The continuum problem
- 2014: Christopher Clapham and James Nicholson: The Concise Oxford Dictionary of Mathematics (5th ed.) ... (previous) ... (next): natural number