Model Theory - Equivalence of formulas using automorphisms

Question

Let $\mathbf Q$ denote the additive group of rational numbers, i.e. the structure $\mathbf Q = (\mathbb Q;+,0)$. Let $L$ be the language of $\mathbf Q$ and let $T$ be the complete theory of $\mathbf Q$. By considering automorphisms of $\mathbf Q$ given that every formula in F1(L) is E1(T)-equivalent to exactly one of the four formulas $v_1=v_1$, $v_1=0$, $v_1\neq 0$, $v_1\neq v_1$.

Prove that there are infinitely many E2(T) - equivalence classes of formulas in F2(T)?

Answer 1

Note that for any $\frac{p}{q}, \frac{p'}{q'}$ with $p,p',q,q'\neq 0$, there exists an automorphism of $\mathbf Q$ which maps $\frac{p}{q}$ to $\frac{p'}{q'}$. Indeed, the map $x\mapsto \frac{qp'}{q'p}x$ is an automorphism of the structure (i.e., it is bijective and preserve $0$ and $+$ in both directions). This means that there are two orbits of points of $\mathbb Q$ under automorphisms of $\mathbf Q$: there is $\{0\}$, and there is $\{\frac{p}{q} \mid p,q\neq 0\}$.

Now, let $\phi(v)$ be a $L$-formula with one free variable. Remark that if $\frac{p}{q}$ satisfies $\phi$, every point $\frac{p'}{q'}$ in the same orbit as $\frac{p}{q}$ also satisfies $\phi$, by the very definition of an automorphism. I can't stress enough how this is important: the set of points $\frac{p}{q}$ for which $\phi$ hold is a union of orbits of $\mathbb Q$ under $\operatorname{Aut}(\mathbf Q)$.

So what kind of formulas with one free variable can we have?

• There are the formulas that are satisfied by no points in $\mathbb Q$, and these ones are equivalent to $v_1\neq v_1$,
• there are the formulas that are satisfied by every point of $\mathbb Q$, which are equivalent to $v_1=v_1$.

These formulas are those that contain either $0$ orbits, or $2$ orbits. It remains those that contain precisely $1$ orbit of points. These ones are:

• The formula that are satisfied only by $0$, and these formulas are equivalent to $v_1=0$,
• and finally, there remains the formulas that are satisfied by the points different than $0$, and those are equivalent to $v_1\neq 0$.