5. 0 is a natural number.
6. For every natural number n, S(n) is a natural number.
Peano's original formulation of the axioms used 1 instead of 0 as the "first" natural number. This choice is arbitrary, as axiom 5 does not endow the constant 0 with any additional properties. However, because 0 is the additive identity in arithmetic, most modern formulations of the Peano axioms start from 0. Axioms 5 and 6 define a unary representation of the natural numbers: the number 1 is S(0), 2 is S(S(0)) (= S(1)), and, in general, any natural number n is Sn(0). The next two axioms define the properties of this representation.
7. For every natural number n, S(n) ≠ 0. That is, there is no natural number whose successor is 0.
8. For all natural numbers m and n, if S(m) = S(n), then m = n. That is, S is an injection.
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Here i find nothing that supports what you did, i.e. using 1 and 3 ambiguously.