Intuitively, one can say that $S(n) > n$. But how do we prove it using the Peano Axioms. It seems like I need a formal statement as to what $>$ means.
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Usually, it's $\leq$ which gets defined first, not $>$. In the case of PA, you can define $\leq$ as $$ a \leq b \leftrightarrow \exists c\: (b = a+ c) $$
But of cource, once you've defined one of the relations $\leq$, $<$, $>$, $\geq$, definitions for the others follow immediately. You e.g. have $$ a > b \leftrightarrow (a \neq b) \land (b \leq a) $$
Or you can define $>$ directly as $$ a > b \leftrightarrow \exists c\: (c \neq 0) \land (a = b + c) $$
Here is a definition that uses the order axiom:
The subset $P$ is the positive numbers of the field. From here, the relation of 'strictly greater than' can be defined as follows.