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  "body": "\n## Summary\n\nThe Peano axioms are axioms for the [[natural numbers|0MWKJ44V5A]].\nThey 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]].\nThe axioms state three properties of these data.\nFirst, $S(n) \\neq 0$ for each $n$ in $\\mathbb{N}$.\nSecond, $S$ is injective: if $S(m) = S(n)$, then $m = n$.\nThird, the induction axiom holds: if a subset $A$ of $\\mathbb{N}$ contains $0$ and contains $S(n)$ for each $n$ in $A$, then $A = \\mathbb{N}$.\n\n## Remarks\n\nGiuseppe Peano published the axioms in 1889. Richard Dedekind gave a similar system in 1888.\nThe 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.\nThe axioms are the base for the [[principle of mathematical induction|0ASJVEFA9J]] and for [[definition by recursion|0DSHAVH45F]].\n\n## Explanations\n\n### L1 human\n\nThe Peano axioms say what is special about counting: you start at $0$ and you always go to the next number.\nTwo different numbers never have the same next number, and you never come back to $0$.\nThe induction axiom says that you reach every natural number in this way.\n",
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