Past exam of the mathematics course of the University of Cambridge / 2021 / iii / Paper 144 / 3

Every model of $T$ becomes an $L_0$-structure by forgetting the symbols outside $L_0$, so $T_0$ has an infinite model. If $\phi$ is an $L_0$-sentence, completeness of $T$ gives either $T\models \phi$ or $T\models \neg \phi$. Accordingly either $\phi \in T_0$ or $\neg \phi \in T_0$. The two cannot both occur because $T$ is consistent. Therefore $T_0$ is a complete $L_0$-theory with infinite models.

Yes. By the Ryll-Nardzewski theorem, aleph-zero-categoricity of $T$ says that, for every $n$, only finitely many $L$-formulas in $n$ variables exist modulo equivalence over $T$. The $L_0$-formulas form a subcollection. Moreover, two $L_0$-formulas are equivalent modulo $T_0$ exactly when their universal equivalence sentence belongs to $T_0$, equivalently when it follows from $T$. Thus there are only finitely many $L_0$-formulas modulo $T_0$ in each arity. Applying Ryll-Nardzewski again proves that $T_0$ is aleph-zero-categorical. This is the reduct of an aleph-zero-categorical theory property.

No. Let $L$ have unary predicates $P,Q$ and a unary function $f$. Let $T$ say that $P$ and $Q$ partition the universe into two infinite sets and that $f$ is an involution mapping $P$ bijectively onto $Q$. This is a complete theory. In every infinite cardinal $\kappa$, a model of total size $\kappa$ has $|P|=|Q|=\kappa$, and any two such bijections are isomorphic. Hence $T$ is $\kappa$-categorical for every infinite $\kappa$.

Take $L_0=\{P,Q\}$. The reduct theory merely says that $P$ and $Q$ are two infinite parts. At any uncountable $\kappa$, it has one model with $|P|={\aleph }_0$ and $|Q|=\kappa$, and another with $|P|=|Q|=\kappa$. They are not isomorphic. Therefore $T_0$ is not $\kappa$-categorical, giving the reduct need not preserve uncountable categoricity counterexample.