> With suitable interpretations of first element and successorship,
That is the crux of the matter! If the first element is zero, and if successor means add one, then the things that the Peano axioms are about are indeed the natural numbers. But the 0 and S of the language of (first order) Peano arithmetic are undefined primitives.
> the > Peano axioms define every and any inductive set, which either form of |N > is certainly one of.
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