Satisfiable formula only over even structares

Question

In First order Logics, what formula can I cook up, that's satisfiable over all even structures, and only even structures. (even structure means it has an even number of elements in its domain).

Answer 1

Let our language have a binary relation symbol $R$. Let $\varphi$ be the sentence which is the conjunction of the following sentences:

(i) the sentence $\forall x\lnot R(x,x)$ that says $R$ is antireflexive;

(ii) the sentence $\forall x\forall y(R(x,y)\to R(y,x))$ that says $R$ is symmetric;

(ii) the sentence $\forall x\exists y(R(x,y)\land \forall t(R(x,t)\to t=y))$.

Then the finite models of $\varphi$ are necessarily of even cardinality. And given any non-empty set $A$ of even cardinality, there is a model of $\varphi$ with underlying set $A$.

If $A$ is an infinite set, then $\varphi$ is satisfiable in $A$. So in that sense all infinite cardinals are even. More precisely, the concepts "even" and "odd" are meaningless for infinite cardinals.