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If we define addition as follows:

  1. Define $a+0=a$.
  2. 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?

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The uniqueness is necessary for addition to be well-defined. For example consider this definition of Smiley-addition.

  1. Define $a+0=a$.
  2. Define $a+1=2$.
  3. For all $a,b\in\mathbb{N}$ such that $a+b$ is defined, define $a+S(b)=S(a+b)$.

You can copy the proof that standard addition defines addition for all pairs of natural numbers to show that the same holds for Smiley-addition. However, you might notice a flaw. Namely, substituting $a=0$ into (2) gives $0+1=2$. But (1) gives $0+0=0$ and (3) tells us that it follows that $0+1=0+S(0)=S(0+0)=1$. We find that the rules of Smiley-addition do not define $a+b$ uniquely, so it is not well-defined, which basically means it is completely useless.

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