Natural Number¶

Definition¶

We define the natural numbers using the Peano axioms. The natural numbers are abstract objects that satisfy the following axioms:

1. The number $0$ is a natural number.
2. For every natural number $n\text{,}$ its successor $S(n)$ is also a natural number.
3. For every natural number $n\text{,}$ the successor $S(n)$ is not $0\text{.}$
4. If $S(n)=S(m)$ for natural numbers $n$ and $m\text{,}$ then $n=m\text{.}$
5. Let $P(n)$ be a statement about a natural number $n\text{.}$ If $P(0)$ is true, and if $P(k)$ implies $P(S(k))$ for every natural number $k\text{,}$ then $P(n)$ is true for every natural number $n\text{.}$

The set of natural numbers is denoted as $\mathbb{N}\text{;}$ informally, $\mathbb{N}=\{0,\,1,\,2,\,3,\ldots\}\text{,}$ and $\mathbb{N}^+$ excludes $0\text{.}$ Once addition is available, the successor $S(n)$ is simply $n+1\text{.}$

Interpretation¶

Natural numbers provide the foundational objects for counting and induction. The Peano axioms do not construct them from earlier number systems; they describe them by a starting point $0$ and a single step, the successor $S$.

The figure shows the chain $0\to1\to2\to3\to4\to\cdots$$\cdots$, each arrow one application of $S(n)=n+1$$1$. No arrow points to $0$ (axiom 3), no number receives two arrows (axiom 4), and every number is reached from $0$ in finitely many steps, so $\mathbb{N}=\{0,S(0),S(S(0)),\ldots\}$ (axiom 5).

Each axiom rules out a different wrong picture. Without axiom 3, the chain could return to $0$, like the hours $0,1,\ldots,11$ on a clock with $S(11)=0$$0$. Without axiom 4, it could run into a loop later: on $\{0,1,2\}$ with $S(0)=1$$1$, $S(1)=2$ and $S(2)=1$$1$, the number $1$ is the successor of two numbers. Without axiom 5, extra elements that are never reached from $0$ could be added.

Axiom 5 is the principle of mathematical induction, and the same start-and-step pattern defines quantities recursively, such as the factorial.

Conventions differ between texts: some start the natural numbers at $1$. On this page $0\in\mathbb{N}$ and $\mathbb{N}^+=\{1,2,3,\ldots\}$$\{1,2,3,\dots \}$; check which convention a source uses before applying a statement about all $n\in\mathbb{N}$$\mathbb{N}$.

Key Results¶

• The Peano axioms characterize the natural numbers: any two systems satisfying them differ only in the names of their elements.
• Axiom 5 is the principle of mathematical induction: every induction proof has a base case and an inductive step.
• Axiom 3 prevents the counting process from cycling back to $0\text{;}$ axiom 4 says two different natural numbers cannot share a successor.
• Under the decimal system, natural numbers include $0,\,1,\,2,\,\dots$

Examples¶

• By induction (axiom 5), $0+1+\cdots+n=\frac{n(n+1)}{2}$ for every $n\in\mathbb{N}\text{:}$ it holds for $n=0\text{,}$ and if it holds for $k\text{,}$ then adding $k+1$ gives $\frac{k(k+1)}{2}+(k+1)=\frac{(k+1)(k+2)}{2}\text{.}$

Prerequisites¶

• Set

Related Concepts¶

• Mathematical Induction
• Integer
• Sequence
• Finite and Infinite Sets

Sources¶

• Math I: Calculus · Lecture 1

Aliases: nonnegative integer Keywords: whole number, counting number, Peano axioms Symbols: element of, natural numbers

Last updated: 20 October 2026 03:02