The set of natural numbers $\N$ is defined as the smallest inductive set that contains $\empt$. See Finite Ordinal.
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
\[\N^+ = \N \setdif \{0\}.\]
$(\N, +, \leq)$ is a well-ordered abelian monoid with neutral element $0$.
$(\N^+, \cdot, \leq)$ is a well-ordered abelian monoid with neutral element $1$.