We often encounter a definition of a function over the natural numbers such as
f (0, m) = 0
f (n + 1, m) = f (n, m) + m
Is this a legitimate definition? That is, is there really a function that satisfies the above two equalities for all n and m? And if so, is there only one such function, or is the definition ambiguous? We address this question in this section through a somewhat more general related question.