Definition:Natural Numbers
Informal Definition
The natural numbers are the counting numbers.
The set of natural numbers is denoted $\N$:
- $\N = \left\{{0, 1, 2, 3, \ldots}\right\}$
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.
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 Elements of Minimal Infinite Successor Set
Let $\omega$ denote the minimal infinite successor set.
The natural numbers can be defined as the elements of $\omega$.
In terms of the empty set $\varnothing$ and successor sets, we 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
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 $\omega$.
Also see
- Definition by Induction of 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
- Bert Mendelson: Introduction to Topology (1962)... (next): $\S 1.1$: Introduction
- 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 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): $\S 1.2$: The set of real numbers
- 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