If we define addition as follows:
- Define $a+0=a$.
- For all $a,b\in\mathbb N$ such that $a+b$ is defined, define $a+S(b)=S(a+b)$.
It's easy to show through induction that this defines addition for all $a,b\in\mathbb N$. A few mathematical papers on Peano axioms such as this one include a proof of the uniqueness of addition. I don't particularly have any problem with that, but I'm curious as to what circumstance needs this uniqueness. For example, will associativity, commutativity and all other properties of addition be proved without having to ensure uniqueness?