Natural numbers - Knowlpedia

The natural numbers form a set $\mathbb N$ with an element $0$ and a successor function $S:\mathbb N\to\mathbb N$ satisfying:

Injectivity: $S(m)=S(n)$ implies $m=n$ .
Zero is not a successor: $S(n)\ne0$ for every $n\in\mathbb N$ .
Induction: every subset $A\subseteq\mathbb N$ containing $0$ and satisfying $S(A)\subseteq A$ equals $\mathbb N$ .

Notation and convention

Write $1=S(0)$ , $2=S(1)$ , and so on, so that $\mathbb N=\{0,1,2,\ldots\}$ . Some authors exclude zero; in this corpus, a positive index is written explicitly as $n\ge1$ . These Peano axioms with induction over all subsets characterize the system up to isomorphism.

Arithmetic

Addition and multiplication are determined recursively by

a+0=a,\qquad a+S(b)=S(a+b),

a\cdot0=0,\qquad a\cdot S(b)=a\cdot b+a.

Remarks

Natural numbers are used to index a sequence and to formulate principles such as mathematical induction and the well-ordering principle. They embed naturally into the integers.

Examples

The numbers $0,1,2,3$ belong to $\mathbb N$ .
The set $\{n\in\mathbb N:n\text{ is even}\}$ is an infinite subset of $\mathbb N$ .