Peano axioms: 3 or 5 axioms?

Question

The Peano axioms are usually stated as follows:

1. Zero is a number.

2. If a is a number, the successor of a is a number.

3. zero is not the successor of a number.

4. Two numbers of which the successors are equal are themselves equal.

5. (induction axiom.) If a set S of numbers contains zero and also the successor of every number in S, then every number is in S.

One can formalize Peano arithmetic as a theory of second-order sentences over the signature $\{0, s\}$ just by formalizing the axioms 3 to 5. So are 1 and 2 really axioms or are they just type declarations?

Answer 1

You're right, in the formalism of second-order logic the first two "axioms" are really the signature.

Of course there might (for all I know) be other formalisms where this way of writing them makes sense.

Answer 2

Peano's original paper had nine axioms, actually. He included axioms such as "for every number $x,$ $x = x$". At the time (1889) there was no well established system of logic, so Peano was starting, essentially, with nothing.

Today, the properties of equality and the fact that function symbols are interpreted by total functions are taken as part of the underlying "logic", apart from the "theory" that is being studied. So axioms 1 and 2 are not needed as axioms. It is necessary, though, to state the signature of the theory. So, if axioms 1 and 2 are left out, it would still be necessary to say that that theory has a constant symbol $0$ and a unary function symbol $s$.