My first idea on how to do this would be:


1. $0\in\mathbf Z$
2. $\forall x\in\mathbf Z,Sx\in\mathbf Z\land Px\in\mathbf Z$
3. $P$ and $S$ are injective
4. $\forall x\in\mathbf Z,PSx=SPx=x$
5. some induction axiom that holds for decreasing sequences as well (intuitively, if $\phi$ holds for $x\_0\in\mathbf Z$ and $\forall x\in\mathbf Z,\phi(x)\to\phi(Px)$ then $\phi$ holds for all numbers less than $x\_0$)
6. addition (with the two standard axioms $\forall x\in\mathbf Z,x+0=x$ and $\forall x,y\in\mathbf Z,Sx+y=S(x+y)$ and an extra axiom $\forall x,y\in\mathbf Z,Px+y=P(x+y)$)
7. multiplication


defining $-1=P0$, $-2=PP0$, etc.


Is this the canonical way to do it (if so, what is it called) or is there a problem I haven't noticed (if so, what would be the "right way")?



What is the canonical way to extend Peano's axioms to the set of all integers?
The canonical way to extend Peano arithmetic to the integers is not by changing the language or axioms, but instead by treating integers as equivalence classes of pairs of natural numbers.


We think of an integer $x$ as any pair $(a,b)$ of natural numbers with $a-b=x$. More formally, we translate arithmetic statements about integers by first replacing the integers with pairs of natural numbers, and then transforming the results by
\begin{align}
0\_Z& \to (0,0)\\
S(a,b)& \to (Sa,b)\\
(a,b)+(c,d)& \to (a+c,b+d)\\
(a,b)(c,d)& \to (ac+bd,ad+bc)\\
(a,b)=(c,d)& \to a+d=b+c
\end{align}


This process turns every arithmetic statement about integers into an arithmetic statement about natural numbers instead. Eg $$(\exists x,y,z\in\mathbb{Z})x^3+y^3+z^3=30$$
can be translated idiomatically (i.e. after performing the mechanical translation and then some of the usual simplifications) as
\begin{align}(\exists a,b,c,d,e,f)& a^3+3ab^2+c^3+3cd^2+e^3+3ef^2=\\
&b^3+3ba^2+d^3+3dc^2+f^3+3fe^2+30\end{align}


