Let $\mathcal{Q}$ denote the additive group of rational numbers, i.e. the structure $\left<\mathbb{Q}; +; 0\right>$. Let $\mathcal{L}$ be the language of $\mathcal{Q}$ and let $T$ be the complete theory of $\mathcal{Q}$. By considering automorphisms of $\mathcal{Q}$ given that every formula in $F_1(\mathcal{L})$ is $E_1(T)$-equivalent to exactly one of the four formulas
- $v_1 = v_1$
- $v_1 = 0$
- $-v_1 = 0$
- $-v_1 = v_1$
prove that there are infinitely many $E_2(T)$ - equivalence classes of formulas in $F_2(T)$?
Fn(L) denotes the set of all L-formulas ϕ with FrVar(ϕ)⊆{v1,...,vn}
En(T) denotes the binary relation on Fn(L) defined by
(ψ ,ϕ)∈En(T)⟺T⊨∀v1,...,vn(ϕ(v1,...,vn)⟺ψ(v1,...,vn))
I have proved that considering automorphisms of $\mathcal{Q}$, every formula is equivalent to exactly one of the above formulas, but cannot prove the last part - infinitely many equivalence classes. Please help.