Addition(+) is defined using the successor function(++) in Peano arithmetic as:
- 0 + m = m
- (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 + m = (0++) + m = (0 + m)++ = m++
- 2 + m = (1++) + m = (1 + m)++ = (m++)++
Thanks very much.