1.5. The naturals

Note:

Informal discussions are inadequate for these notes and are thus mostly omitted. For this reason, I assume the following and provide no proof in these notes.

The algebraic structure of $\mathbb{N}$ may be summarized as follows.

Axiom ( $\mathbb{N}$ is a commutative monoid).

If $n,m$ and $k$ are natural numbers, then the addition satisfies the following.

1. $(n+m)+k=n+(m+k)$ (Associative);
2. $n+m=m+n$ (Commutative);
3. if $n+m=n+k$, then $m=k$ (Cancellation law).

Multiplication satisfies the same properties and the two operations interact with each other following the distributive laws: $n\cdot (m+k)=n\cdot m+n\cdot k$.

The order structure of $\mathbb{N}$ can be defined in terms of the algebra as follows.

Definition.

I write $n\le m$ to abbreviate the fact that $n+k=m$ for some $k$. $n<m$ if $n\le m$ but $n$ and $m$ are different naturals.

Reflect.

If I had not introduced the algebra as an axiom, but instead began with the order, could you define the algebraic structure based on the order? Why do you think I chose to do it this way instead?