My textbook (Amann and Escher, Analysis I) gives a theorem which says that the operations of addition and multiplication (and a partial order $\leq$) exist and are uniquely defined by a whole host of characterizations. My question here is in particular about the claim of uniqueness with respect to addition.
The claim with respect to uniqueness is that addition is the unique operation on $\mathbb{N}$ which obeys
Addition is associative, commutative and has the identity element 0.
We note that $0$ has already been defined as the privileged 0 element in $\mathbb{N}$ (i.e. the middle element in the triple $(\mathbb{N},0,\nu)$, where $\nu$ is the successor function).
The way the text seems to prove uniqueness is as follows. It first shows that if two operations $\circledast, \circledcirc$ obey the following three conditions, then they are the same (incidentally, doesn't C2 imply C3 given the $\nu(0) =: 1$ identification?):
(C1) $0 \circledast 0 = 0$
(C2) $n \circledast 1 = \nu(n)$
(C3) $n \circledast \nu(m) = \nu(n \circledast m)$
We then show that $+$ obeys the (Ci), so that any other operation obeying the (Ci) is the same as +. But the authors just leave it here! Apparently the suggestion is that the (Ci) are sufficient and necessary for any operation obeying the characterization of $+$ in the theorem. Sufficiency was shown, but I am not sure to show necessity. Why should commutativity, associativity, and 0 as the identity for a given operation imply the (Ci)?
I here attach the proof in its full gory detail in case my question is not clear (I apologize for the length, so please feel free to ignore -- hopefully my question in bold above is clear!)
