---
tag: 0QFT7Z76N5
kind: concept
name: Peano axioms
aliases: [dedekind-peano-axioms, peano-postulates]
msc: [03F30]
sources:
  - url: https://en.wikipedia.org/wiki/Peano_axioms
    locator: Historical second-order formulation
same_as:
  wikidata: Q842755
  wikipedia: Peano axioms
---

## Summary

The Peano axioms are axioms for the [[natural numbers|0MWKJ44V5A]].
They use a set $\mathbb{N}$, an element $0$ of $\mathbb{N}$ and a function $S \colon \mathbb{N} \to \mathbb{N}$, the [[successor function|0AK9TGSNXY]].
The axioms state three properties of these data.
First, $S(n) \neq 0$ for each $n$ in $\mathbb{N}$.
Second, $S$ is injective: if $S(m) = S(n)$, then $m = n$.
Third, the induction axiom holds: if a subset $A$ of $\mathbb{N}$ contains $0$ and contains $S(n)$ for each $n$ in $A$, then $A = \mathbb{N}$.

## Remarks

Giuseppe Peano published the axioms in 1889. Richard Dedekind gave a similar system in 1888.
The induction axiom speaks about all subsets of $\mathbb{N}$, thus it is a second-order axiom. First-order Peano arithmetic replaces it with an axiom schema.
The axioms are the base for the [[principle of mathematical induction|0ASJVEFA9J]] and for [[definition by recursion|0DSHAVH45F]].

## Explanations

### L1 human

The Peano axioms say what is special about counting: you start at $0$ and you always go to the next number.
Two different numbers never have the same next number, and you never come back to $0$.
The induction axiom says that you reach every natural number in this way.