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 give
$$
\pi(x)=\{\pi(y):y\in^M x\}=x.
$$
Hence $\pi$ is the <identity function>. This is <Paris models are rigid when their complete theory has a well-founded model>, so \b[every Paris model of $T$ is a rigid first-order structure.]

Solution

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

{c}

Under the hypothesis of <Paris models are well-founded when their complete theory has a well-founded model>, a Paris model $M$ is well-founded. Every automorphism of $M$ is then the identity by external <epsilon induction>: once all members of $x$ are fixed, extensionality gives $\pi(x)=\{\pi(y):y\in^M x\}=x$. Hence $M$ is a <rigid first-order structure>.

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 Incoming links: Paris models are rigid when their complete theory has a well-founded model