---
tag: 0D752BG155
kind: statement
role: axiom
name: Zero is not a successor
msc: [03F30]
uses: [0832XVR512, 0QFT7Z76N5]
sources:
  - url: https://en.wikipedia.org/wiki/Peano_axioms
    locator: Historical second-order formulation
---

## Statement

Let $S \colon \mathbb{N} \to \mathbb{N}$ be the [[successor function|0832XVR512]] of the natural numbers.
For each natural number $n$, $S(n) \neq 0$.

## Remarks

This is one of the [[Peano axioms|0QFT7Z76N5]]. In a construction of $\mathbb{N}$, it is a theorem.
With the [[injectivity of the successor function|0TBYMWPWWB]], it shows that the numbers $0, S(0), S(S(0)), \dots$ are all different.

## Explanations

### L1 human

When you count up from any natural number, you never get back to $0$.
Thus the natural numbers do not go around in a circle, like the hours on a clock do.