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


Solution

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

 Solution (source code)