TITLE: Is there a formal definition of "Greater Than"
QUESTION [5 upvotes]: 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.

REPLY [4 votes]: Here is a definition that uses the order axiom:

For an ordered field $\mathbb{F}$, there is a unique subset $P$ satisfying the following conditions.

if $a,b\in{P}$ then $a+b,ab\in{P}$.
for all $a$ in $\mathbb{F}$, one and only one of the following is true: $a\in{P}$, $-a\in{P}$, or $a=0$.

We say that $p \in \mathbb{F}$ is positive iff $p \in P$.

The subset $P$ is the positive numbers of the field. From here, the relation of 'strictly greater than' can be defined as follows.
$$a>b\iff{a-b}\in{P}$$