Definition 1. The set of natural numbers $\N$ is defined as the smallest $\empt$-inductive set. See Finite Ordinal.
Definition 2. The arithmetic on $\N$ is as defined for ordinals (see Ordinal Arithmetic), but without any concern for limit ordinals.
Definition 3. As defined for ordinals, the following gives a well-order on $\N$:
\[n < m \enspace\lrimp\enspace n \in m.\]Definition 4. The set of positive natural numbers is
\[\N^+ = \N \setdif \{0\}.\]Proposition 1. $(\N, +, 0, \leq)$ is a well-ordered abelian monoid.
Proposition 2. $(\N, \cdot, 1, \leq)$ is a well-ordered abelian monoid.