Definition:Ordering on Natural Numbers

Informal Definition
Let $\N$ denote the natural numbers.

The ordering on $\N$ is the relation $\le$ everyone is familiar with.

For example, we use it when we say:


 * James has six apples, which is more than Mary, who has four.

which can be symbolised as:


 * $6 \ge 4$

Every attempt to describe the natural numbers via suitable axioms should reproduce the intuitive behaviour of $\le$.

The same holds for any construction of $\N$ in an ambient theory.

Also defined as
As seen above, not all sources define both the strict ordering $<$ and the weak ordering $\le$ on $\N$.

However, by Reflexive Reduction of Ordering is Strict Ordering and Reflexive Closure of Strict Ordering is Ordering, this is seen to be immaterial.