The set of natural numbers $\N$ is defined as the smallest inductive set (see Natural Numbers (Set Theory)).
The arithmetic on $\N$ is as defined for ordinals (see Ordinal Arithmetic), but without any concern for limit ordinals.
As defined for ordinals, the following gives a well-order on $\N$:
\[n<m \enspace\lrimp\enspace n\in m.\]The set of positive natural numbers is
\[\N^+ = \N\setminus\{0\}.\]
$(\N,+)$ is an abelian monoid with neutral element $0$.
$(\N^+,\cdot)$ is an abelian monoid with neutral element $1$.