According to the Wikipedia entry on the Peano axioms: "the number 1 can be defined as $S(0)$, 2 as $S(S(0))$ (which is also $S(1)$), and, in general, any natural number n as the result of n-fold application of $S$ to $0$, denoted as $S^n(0)$." (where $S(n)$ is the successor function).
The issue I have is with the statement "n-fold application". With these axioms, we are trying to define what the natural numbers are axiomatically. Therefore, we cannot use the natural numbers to define themselves (or so I think). However, using something like "n-fold application" within the axioms--where n is a natural number--is doing precisely that, is it not (using numbers to define what numbers are)?