Successor function

Definition

Let $\mathbb{N}$ be the set of natural numbers. The successor function is the function $S \colon \mathbb{N} \to \mathbb{N}$ with $S(n) = n + 1$ for each $n$ in $\mathbb{N}$. The number $S(n)$ is the successor of $n$.

Remarks

In the Peano axioms, $S$ is a primitive function, and the axioms do not define it. Then $n + 1$ is a name for $S(n)$, and $1$ is a name for $S(0)$. For example, $S(0) = 1$, $S(1) = 2$ and $S(S(2)) = 4$.

Explanations

L1 human

The successor of a number is the number that comes after it when you count. The successor of $4$ is $5$, and the successor of $0$ is $1$.