Once more on countably categorical sentences (Q1307187)

Countably categorical sentences of an infinitary logic are considered in the article under review. A set \({\mathbf{A}}\) is said to be locally countable if \({\mathbf{A}}\vDash\;\forall a\neq\emptyset\) \(\exists f\) \((f\:\omega\overset\text{onto}\rightarrow\omega)\). Let \(\mathbf{A}\) be a locally countable admissible set, \({\mathbf{A}}>\omega\). It is proven that a sentence \(\varphi\) in a fragment \(L\in{\mathbf{A}}\) is countably categorical if and only if all the types realized in models \({\mathcal{M}}\in{\mathbf{A}}\) of \(\varphi\) are principal in \(L\) whenever \(\varphi\) is equivalent to a canonical Scott sentence. The author constructs an example of a sentence that is not countably categorical but only with a hyperarithmetic model. It is proven that the countable models of countably categorical theories which belong to \({\mathbf{A}}\) are exactly all the models in \({\mathbf{A}}\) whose Scott rank is less than \(o(\mathbf{A})\).