Order on the natural numbers

For $m,n\in \mathbb{N}$, define the order by

$m\le n⟺\mathrm{\exists }k\in \mathbb{N}(m+k=n),$

and the strict order by $m<n⟺(m\le n\text{ and }m\mathrm{\ne }n)$, using addition (Addition of natural numbers).

Addition of natural numbers

Read $m\le n$ as "$n$ is $m$ plus something": the gap $k$ with $m+k=n$ is unique when it exists (by commutativity and cancellation, Addition is commutative, Addition is cancellative) and is written $n-m$ once subtraction is available. That $\le$ is genuinely a linear order, reflexive, antisymmetric, transitive, and total, is proved in $\le$ is a linear order on $\mathbb{N}$ and Trichotomy of the order on $\mathbb{N}$; compatibility with the arithmetic (Order is compatible with addition, Order is compatible with multiplication) and discreteness ($m<n⟺\sigma (m)\le n$, Discreteness: $\sigma (n)$ is the immediate successor) follow.

In the von Neumann model this order coincides with membership and inclusion: $m\le n⟺m\subseteq n$ and $m<n⟺m\in n$, matching the picture of each natural number as the set $\{0,1,\dots ,n-1\}$ of its predecessors (The natural numbers $\mathbb{N}$ (von Neumann)). That coincidence is not proved here; it is derived from the additive definition above on a later page. The least-element property of every nonempty subset, the well-ordering principle (The well-ordering principle), is the deepest consequence.