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#1: Initial revision
You could use the Peano axioms instead to define addition - using the successor function, $$a + 0 = a $$ $$a + S(b) = S(a + b)$$ But I'd argue that this is not different - addition is defined this way to begin with... In my opinion the question makes no sense - addition is addition, you can break it down to its definition, but that's still addition...