Order on the natural numbers

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

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

and the strict order by $m < n \iff (m \le n \text{ and } m \neq n)$, using addition (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 ($m < n \iff \sigma(m) \le n$, Discreteness: $\sigma(n)$ is the immediate successor) follow.

In the von Neumann model this order coincides with membership and inclusion: $m \le n \iff m \subseteq n$ and $m < n \iff m \in n$, 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.