Peano system

A Peano system is a triple $(N,0,\sigma )$ consisting of a set $N$, a distinguished element $0\in N$, and a successor function $\sigma :N\to N$ (A function is a relation $f$ with $(a,b)\in f$ and $(a,c)\in f$ implying $b=c$; $f:A\to B$, the value $f(a)$, domain and codomain), satisfying the three Peano axioms:

• (P1) Zero is not a successor. For every $n\in N$, $\sigma (n)\mathrm{\ne }0$.
• (P2) Successor is injective. For all $m,n\in N$, if $\sigma (m)=\sigma (n)$ then $m=n$.
• (P3) Induction. For every subset $S\subseteq N$ (Subset $x\subseteq y$, proper subset $x⊊y$, and the separation notation $\{z\in x:\phi (z)\}$), if $0\in S$ and $\sigma (n)\in S$ whenever $n\in S$, then $S=N$.

The elements of $N$ are the natural numbers of the system. We abbreviate $1:=\sigma (0)$, $2:=\sigma (1)$, $3:=\sigma (2)$, and so on.

Axiom (P3), the induction axiom, is the load-bearing one: it says $N$ has no proper subset that contains $0$ and is closed under $\sigma$, so every element is reached from $0$ by finitely many successors. It is what makes definition by recursion and proof by induction legitimate (The principle of mathematical induction, The recursion theorem).

This is a structural definition: it does not say what the natural numbers are, only which laws they obey. Two questions are therefore separate and both require proof: that a Peano system exists (a model built inside set theory, The von Neumann naturals form a Peano system, from the von Neumann naturals The natural numbers $\mathbb{N}$ (von Neumann)), and that it is unique up to a unique isomorphism (Categoricity: the natural numbers are unique up to unique isomorphism). Together these justify speaking of the natural numbers $\mathbb{N}$.

We use the modern $0$-based convention ($0\in N$). Peano's original postulates started from $1$; the choice is immaterial, and starting at $0$ matches the set-theoretic construction, where $0=\mathrm{\varnothing }$.