The only remaining problem is the unlikely situation that S(a) doesn't just return a result but affects the original value of a.

I'm assuming this as a maths problem not a computation problem, so a function maps one value onto another (OK, maybe it's "a set of ordered pairs such that ...") - the question of changing the "original value" doesn't arise.

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.

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.

I got lazy and also distracted by the next

When and if I ever come back to it, I will try to remember to give it a fresh try before looking it up.

Quote:

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.

You're on to me! Yes, I had seen a video about Peano axioms and natural numbers. I wanted to see if I derive some of the proofs myself, given the same axioms. (axia?)