“if you don’t assume it, then you can’t prove that all natural numbers can be reached by repeatedly applying S(x) from 0.” But why can’t you just assume that as an axiom, instead of induction?
How would you say “repeatedly” in formal language? Would it be something like, ‘every number can be reached by applying the successor operation some number of times’? To the extent we can say that, it already seems included in the definition of addition. But it doesn’t rule out non-standard numbers, because you can reach one of them from zero by applying S(x) a non-standard number of times.
IIRC some of the axioms use recursion so I don’t see why that wouldn’t be allowed. I’m not entirely certain how you would set it up but it doesn’t seem like it should be impossible. That link looks interesting though, perhaps it addresses it.
How would you say “repeatedly” in formal language? Would it be something like, ‘every number can be reached by applying the successor operation some number of times’? To the extent we can say that, it already seems included in the definition of addition. But it doesn’t rule out non-standard numbers, because you can reach one of them from zero by applying S(x) a non-standard number of times.
IIRC some of the axioms use recursion so I don’t see why that wouldn’t be allowed. I’m not entirely certain how you would set it up but it doesn’t seem like it should be impossible. That link looks interesting though, perhaps it addresses it.