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 $n \in N$, $\sigma(n) \neq 0$.
• (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 $x \subsetneq y$, and the separation notation $\{\, z \in x : \varphi(z) \,\}$), 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 $0 = \varnothing$.