Is there a formal definition of "Greater Than"

Question

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.

Answer 1

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 $$ a > b \leftrightarrow \exists c\: (c \neq 0) \land (a = b + c) $$