My query is regarding following question:-
Let $\mathcal Q$ denote the additive group of rational numbers, i.e. the structure $\langle Q ; +; 0\rangle$. Let $\mathcal L$ be the language of $\mathcal Q$ and let $\mathcal T$ be the complete theory of $\mathcal Q$.
(i) By considering automophisms of $\mathcal Q$ prove that every formula in $F_1(\mathcal L)$ is $E_1(\mathcal T)$-equivalent to exactly one of the four formulas $v_1\bumpeq v_1; v_1\bumpeq 0; \neg v_1\bumpeq 0; \neg v_1\bumpeq v_1$:
(ii) Deduce that there are exactly two 1-types (over T), both of which are principal.
(iii) Show that there exists a 2-type (over T) which is not realised in $\mathcal Q$ and deduce that T is not $\aleph_0$-categorical.
$F_n(\mathcal L)$ denotes the set of all $\mathcal L$-formulas $\phi$ with FrVar($\phi$)$ \subseteq \{v_1,...,v_n\}$
$E_n(\mathcal T)$ denotes the binary relation on $F_n(\mathcal L)$ defined by
($\psi\space,\phi)\in E_n(\mathcal T)\iff\mathcal T\models\forall v_1,...,v_n (\phi(v_1,...,v_n)\iff\psi(v_1,...,v_n))$
I am wondering if someone could help me understand and solve this question.
(i) The automorphism is $\pi :\mathcal Q \rightarrow \mathcal Q $ such that for $r\in Q $ we have $\pi(r)=r\space\pi(1)$. I just don't know how automorphism is used to deduce that "every formula in $F_1(\mathcal L)$ is $E_1(\mathcal T)$-equivalent to exactly one of the four formulas $v_1\bumpeq v_1; v_1\bumpeq 0; \neg v_1\bumpeq 0; \neg v_1\bumpeq v_1$".
Similar question is solved in course notes without description. Only automorphism is given and then without explanation lecturer says, " From this automorphism we get four formulas...." Can someone please explain the role of automorphism here?
(ii) I deduced that $v_1\bumpeq 0$ and $\neg v_1\bumpeq 0$ are two principal formulas that generates two principal 1-type $\{v_1\bumpeq 0,v_1\bumpeq v_1\}$ and $\{\neg v_1\bumpeq 0,\neg v_1\bumpeq v_1\}$.
I deduced so because $\mathcal T\models\forall v_1( v_1\bumpeq 0\implies v_1\bumpeq v_1)$ , $\mathcal T\models\forall v_1( v_1\bumpeq 0\implies v_1\bumpeq 0)$ and $\mathcal T\models\forall v_1(\neg v_1\bumpeq 0\implies \neg v_1\bumpeq v_1)$ ,$\mathcal T\models\forall v_1( \neg v_1\bumpeq 0\implies \neg v_1\bumpeq 0)$
Am I right?
(iii) I am confused how to solve this last part. I am thinking along this way. Please let me know if I am thinking along the right line. If I give example that show there are infinitely many $E_2(\mathcal T)$-equivalence classes in $F_2(\mathcal L)$ then I deduce there exists a non-principal 2-type say $p$ (by theorem in notes).Then by 'Omitting types theorem' I can deduce that there exists a countably infinite model of $\mathcal T$ that omits $p$. By another theorem in notes I know that there exists countably infinite model that realizes $p$. Clearly both these models are not isomorphic otherwise they would be realising same 2-types. So $\mathcal T$ is not $\aleph_0$- categorical.
Please ask me if you need any further information.