Correct. All descriptions of the natural numbers of which I'm aware require the assumption that one can take some object and apply some operation 'S' on that ob

Correct. All descriptions of the natural numbers of which I'm aware require the assumption that one can take some object and apply some operation 'S' on that object to produce another object that is neither the original object nor any of the objects produced before by repeated applications of 'S' to some root object. Whether you simply define a number as 0 and S(x) where x is a natural number or you define natural numbers as nested sets in ZFC or as church-encoded lambda terms, etc, you assume the existence of such a transformation.