---
tag: 0832XVR512
kind: definition
name: Successor function
msc: [03F30]
defines: 0AK9TGSNXY
uses: [0MWKJ44V5A, 0QFT7Z76N5]
sources:
  - url: https://en.wikipedia.org/wiki/Successor_function
    locator: lead section
  - url: https://en.wikipedia.org/wiki/Successor_function
    locator: Overview
---

## Definition

Let $\mathbb{N}$ be the set of [[natural numbers|0MWKJ44V5A]].
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|0QFT7Z76N5]], $S$ 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 $1$.