How did you go with this?
I was lazy and after a little bit I looked up some proof sketches rather than figuring it out for myself.
These are two of the Peano axioms for the natural numbers. Did you have any other axioms (not counting first-order logic)? Because without some of the other Peano axioms it seems like you can't rule out models other than the natural numbers - for example you need to say there is no x such that S(x) = 0 otherwise negative integers are OK. Likewise the axiom of induction (all elements can be reached by iterating the successor function) is needed to exclude the real. Of course the integers and the reals are commutative, but you can't prove it by induction. At least not as easily.