For a system to be Peano's you requite 3 things: 1) First element isn't a succesor of any other element 2) "Successor" function is injective. 3) If $A \subseteq P$, First element is in $A$, and $S(A)\subseteq A \Rightarrow A=P$
But i've got a sucesor function which makes impossible to find this $A$ set. The fucntion is $S:P\rightarrow P$.
$S(a) = -a$ with $a>0$
$S(a)=-(a-1)$ with $a<0$
$S(a)=1$ for a=0
P is the set of integers, also the first element of my system is $0\in P$ (not $1$)