If we remove the axiom that zero doesn't have a predecessor, and stipulate that every natural number has a predecessor, and that no number is the successor of itself. And keep all other axioms of $\small \sf PA$, and introduce a new primitive relation symbol $``<"$ to signify strictly smaller than relation, and axiomatize that each $n^{th}$ successor of zero [for a meta-theoretic $n$] is strictly smaller than its successor, and that $<$ is asymmetric and transitive.
Formally this is the first order theory with axioms of:
axioms of first order $\small \sf PA$ minus $\not \exists x (S(x)=0)$ plus:
- $\forall x \exists y: S(y)=x$
- $S(x) \neq x$
Define recursively: $S_0(0) = 0 \\ S_{i+1}(0)=S(S_i(0))$
For $n=1,2,3,....\\ S_n(0) < S_{n+1}(0)$
$x < y \to \neg \ y < x$
$ x < y < z \to x < z$
This is what I term as $\small \sf cyclic \ PA$ or $\small \sf cPA$
Question: is there an interpretation of $\small \sf cPA$ in $\small \sf PA$?