Peano's second axiom.

Hi all,

How is exactly Peano's second axiom writtin in FOL?

The axiom reads: The successor of a number is a number.

Is it: AxAy( (x is a number & y=S(x)) -> y is a number ).

Or is it:

Ax( x is a number -> S(x) is a number ).

The reason beyond this question is that, if it is the first formula,
then Peano's axioms are incomplete, because we would need the axiom

Ax( x is a number -> Ey( y=S(x) ) ).

Zuhair

.

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