\begin{definition}[Definition:Ordering on Natural Numbers/1-Based]
Let $\N_{>0}$ be the axiomatised $1$-based natural numbers.
The '''strict ordering of $\N_{>0}$''', denoted $<$, is defined as follows:
:$\forall a, b \in \N_{>0}: a < b \iff \exists c \in \N_{>0}: a + c = b$
The '''(weak) ordering of $\N_{>0}$''', denoted $\le$, is defined as:
:$\forall a, b \in \N_{>0}: a \le b \iff a = b \lor a < b$
\end{definition}