Variations in the successor function from Peano's axioms

It works ...

We have to note that according to Peano's axioms the symbol $1$ is introduced as the name for the successor of $0$; thus, according to your approach, we have to use $1$ to denote the "real" number $2$ (the "new" successor of $0$).

Consider that Peano's axioms include the recursive axioms for sum and product.

For sum :

$\forall x(x+0=x)$

$\forall x \forall y (x+S(y)=S(x+y))$.

Thus, applying the second axiom with e.g. $x :=1$ and $y := 0$, we have that :

$1+S(0)=S(1)=4$,

because in the "reformed" sequence : $0,2,1,4,3,\ldots$, the successor of $1$ is $4$.

Consider now :

$2+S(0)=S(0)+S(0)$,

because $2$ is the "new" successor of $0$; thus :

$2+S(0)=S(0)+S(0)=S(S(0)+0)=S(S(0))=S(2)=1$,

because $1$ is the successor of $2$.

Thus, your proposed "reform" seems consistent, because the operation of "adding to $n$ the successor of $0$" still produces the successor of $n$.

Now, there is an easy way to avoid this apparent confusion : we can introduce new symbols :

$1^*$ for the successor of $0$, i.e. as a new "name" for $2$,

$2^*$ for the successor of the successor of $0$, i.e. for the successor of $2$, i.e. as a new "name" for $1$,

$3^*$ for the successor of the successor of the successor of $0$, i.e. for the successor of $1$, i.e. as a new "name" for $4$,

and so on.

In this way, we have relabeleld the "reformed" sequence : $0,2,1,4,3,\ldots$ with a sequence of new "names" : $0, 1^*, 2^*,3^*,\ldots$.

Using them in the above "computations", we get :

$2^*+S(0)=2^* + 1^*=3^*$

and it is correct, because $3^*$ is the new "name" for $4$.

In the same way :

$1^*+S(0)=1^*+1^*=2^*$

and again it is correct, because $2^*$ is the new "name" for $1$.

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Conclusion : the proposed "reform" us useless.

According to Peano's axioms the "real" number $1$ has only one "relevant" property : to be the (unique) successor of $0$, i.e. the successor of the unique number without successor.

As long as we satisfy the two basic property of the successor function :

• $0$ has no successor

• no two different numbers can have the same successor

there are no "metaphysical" properties that can distinguish two numbers for each other if not their "relative position" with respect to $0$.