In the thread Even Xor Odd Infinities I defined odd models of Modular Arithmetic (MA) as models satisfying the axioms of MA and two first order statements. Even XOR Odd Infinities?
$O1) \forall x(x=0 \lor x+x \ne 0)$
$O2) \exists x(x+x=S0)$
MA has the same axioms as first order Peano Arithmetic (PA) except $\forall x(Sx \ne 0)$ is replaced with $\exists x(Sx=0)$.
Ashutosh proved $O2 \to O1$. Assume $x+x=0$ and $y+y=1$.
Ashutosh's proof doesn't use induction or the axiom $\exists x(Sx=0)$. This means it works even in theories weaker than PA and MA. Emil Jerabek proved $O1$ does not imply $O2$ in MA by proving the 2-adic numbers are a counter example.
The standard models of MA are the rings $Z/nZ$. Any infinite model of MA must be non-standard. Several people have stated any algebraically closed field is a model of MA. If so, Ashutosh's proof shows algebraically closed fields are odd models of MA. The complex numbers satisfy every definition for odd I have been able to come up with. This seems to suggest there are an odd number of complex numbers.
I am looking for first order arithmetic statements true in rings $Z/nZ$ if and only if n is odd. Showing how the statement implies or is implied by other definitions of odd would be an added bonus. Finding a statement independent of others would be interesting. Finding a definition of oddness that is not true in the complex numbers would be even more interesting. Some examples:
$O3) \exists x(x+Sx=0)$
$O4) \forall x \exists y(x=y+y)$
$O5) \forall x \forall y(x=0 \lor y=0 \lor Sx \ne 0 \lor x*y \ne y)$
$O5$ is a complicated way of saying $\forall x(-x \ne x)$
All of these statements are independent of the axioms of MA and require the axiom $\forall x(Sx \ne 0)$ to prove (or disprove) in PA.