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Peano arithmetic

Peano's axioms, successor, axiom of induction
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Mathematics Subject Classification

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Peano originally didn't use 0 as the start of the Natural numbers. He used 1 in axioms 1 and 3. The use of 0 as the beginning of the "natural" numbers is a set theoretic concept, and as such is not "natural" in the least. The idea of zero as a number took ages to develop, and is anything but natural.

Also, in Peano Arithmetic, you have to define 1 as the successor of 0 before you can use it and there is no justification from the axioms for using it in the definitions of addition and multiplication rather than something else. But by starting with 1 as the first natural number, the definitions follows from the axioms as 1 generates all the natural numbers by successive addition, which is something 0 doesn't do.

I would prefer that you call the Axioms in this section The Extended Paeno Axioms, since these are not what he wrote in his papers.

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