Lets consider $(A,g,a_{0})$ , where $A$ is a set, $g$ is a function and $a_{0}$ is the first element of $A$. If $(A,g,a_{0})$ follows all the Peano's axioms, than we can find an isomorphism f between $(A,g,a_{0})$ and $(N,s,0)$. N is the set of natural numbers and s is the Peano s function. $f(a_{0})=0$ and $(f•g)(a)=(s•f)(a)$
My question is: shall g be equal to s? I will give an example
Lets consider A={1,3,5,7,9,11,...} the set of odd numbers. Can i define g(1)=3, g(3)=5,...,Etc.?
It exists an isomorphism f between A and N. Can i say that $f:(A,g,a_{0}) \rightarrow(N,s,0)$ is an isomorphism of Peano's systems?