Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 121 / 1 / iv / b / Solution

By part (a), the Paris model $M$ is well-founded. Let $\pi$ be an automorphism of a first-order structure of $(M,{\in }^M)$. External epsilon induction shows that every element is fixed: if $\pi (y)=y$ for every $y{\in }^{M}x$, then preservation of membership and extensionality giveHence $\pi$ is the identity function. This is Paris models are rigid when their complete theory has a well-founded model, so every Paris model of $T$ is a rigid first-order structure.