Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 2 / 16H / e / i / Solution

A model of $T_G$ consists of a model $M\models T$ and automorphismssuch that ${\vartheta }_g{\vartheta }_h={\vartheta }_{gh}$ and ${\vartheta }_g\ne {\vartheta }_h$ whenever $g\ne h$. Thus $g\mapsto {\vartheta }_g$ is an injective homomorphism $G\hookrightarrow \mathrm{Aut}(M)$. Conversely, any such embedding supplies interpretations of the $f_g$ and hence a model of $T_G$.

Therefore $G$ is $T$-good exactly when $T_G$ has a model. By part (b), this is equivalent to consistency of $T_G$, and hence