Order on the natural numbers

Definition

Let $a$ and $b$ be natural numbers. The notation $a \le b$ means that there is a natural number $c$ with $a + c = b$, where $+$ is the addition of $\mathbb{N}$. The notation $a < b$ means that $a \le b$ and $a \neq b$.

Remarks

The relation $\le$ is a total order on $\mathbb{N}$: it is reflexive, antisymmetric and transitive, and $a \le b$ or $b \le a$ for all $a$ and $b$. The order agrees with the operations: if $a \le b$, then $a + c \le b + c$ and $a c \le b c$ for each natural number $c$. Each nonempty set of natural numbers has a least element for this order. This is the well-ordering principle.

Explanations

L1 human

The number $a$ is at most $b$ when you can get from $a$ to $b$ by adding something. For example, $3 \le 7$ because $3 + 4 = 7$, but $7 \le 3$ is false because no natural number added to $7$ gives $3$.