Peano arithmetic - Why is adding n to m the same as incrementing m n times?

Question

Addition(+) is defined using the successor function(++) in Peano arithmetic as:

1. 0 + m = m
2. (n++) + m = (n + m)++

While these are intuitive axioms that are consistent with my previous, elementary, understanding of addition, I don't understand how it follows from these axioms that n + m is the same thing as incrementing m n times.

Although I can see that it is true for specific cases:

1. 1 + m = (0++) + m = (0 + m)++ = m++
2. 2 + m = (1++) + m = (1 + m)++ = (m++)++
3. etc.

Thanks very much.

Answer 1

We can prove it by induction on $m$. If $m=0$, $n+m=n+0=n$ is the result of not incrementing $n$ at all, i.e. doing it $0$ times. Suppose $n+k$ is the result of $k$ increments starting from $n$. For the inductive step, associativity of $+$ gives $n+(k+1)=(n+k)+1$, i.e. we increment $k$ times, then once more.