I am referring to section 4.1 of Shorter Model Theory.
- Here is how Hodges states and prves Kueker-Reyes theorem:
Hodges says (d) to (a) is clear, but sadly not to me. Can someone explain a bit further?
- Moreover, some pages before, he makes the following statement:
I am not really seeing why we are throwing away information in this process: when we see $Aut(A)$ as a topological group, we are atually putting a topology on $Sym(dom(A))$ so it would seem to me we are adding information.
Thanks in advance for any help
For 1), suppose that there is a tuple $\overline{a}$ such that (d) holds. Let $\alpha$ be in $G_{(\overline a)}$, i.e., $\alpha$ is an automorphism of $B$ such that $\alpha(\overline a)=\overline a$. To prove that $\alpha$ is an automorphism of $A$, it suffices to show that for every atomic $L^+$-formula $\phi(\overline x)$, $A\models\phi(\overline b)$ holds iff $A\models\phi(\alpha(\overline b))$ holds. By assumption, for every $\phi$ there is an $L^-$-formula $\psi$ such that $\phi(\overline b)$ is equivalent to $\psi(\overline b,\overline a)$ for every $\overline b$ in A. Note that automorphisms preserve $L_{\omega_1,\omega}$-formulas, which can be proved by induction (but should be understood on a more "fundamental" level: logic cannot tell apart things that are related by an automorphism). Suppose that $A\models\phi(\overline b)$, so that we have $B\models\psi(\overline b,\overline a)$. Since $\alpha$ is an automorphism of $B$ fixing $\overline a$, we have $B\models\psi(\alpha(\overline b),\overline a)$, so that $A\models\phi(\alpha(\overline b))$. The reverse direction is exactly the same.
Concerning your second question, the way to see how much "information" an object has is to define what we consider to be a valid "isomorphism". If we consider $Aut(A)$ as a permutation group, the notion of isomorphism is that of isomorphism of group actions, i.e., if $H$ is a permutation group acting on a set $B$ then an isomorphism between $Aut(A)$ and $H$ is a bijection $\phi\colon A\to B$ that commutes with the action. If we consider $Aut(A)$ as a topological (abstract) group, then an isomorphism is a group homomorphism that is also a homeomorphism. In a way, we have forgotten that Aut(A) acts on a set, and we only remember the abstract structure of the group together with the topology inherited from the action.
This way, you can see that if $Aut(A)$ and $Aut(B)$ are isomorphic as permutation group, they are isomorphic as topological groups. But the converse is not necessarily true, this is why we are "losing" information.