Order of the natural numbers

Question

The set of natural numbers as given from the Peano axioms $(N,S)$ has an order.

I saw in wikipedia that $(N,+)$ is a commutative monoid, but since the naturals have an order structure by construction shouldn't $(N,+)$ be an ordered commutative monoid ?

Thanks

Answer 1

You can have a square and call it a rectangle. There is no inherent problem with that, as long as all you really need from your quadrilateral is that all angles are right.

In the exact same way, you can take an ordered, commutative monoid and call it a commutative monoid.

Answer 2

$(N,+)$ is both a commutative monoid and an ordered commutative monoid.

Answer 3

Yes, but you can discuss addition without discussing order.

The same way that you can say that $(\mathbb R,+)$ is a group. It is also an abelian group. It is a ring. It is a field. It is an ordered field. It is a module over $\mathbb Z$. Etc., etc.