= Solution
{c}
Take the formula saying that a set is transitive:
$$
\boxed{\varphi_1(x):\quad\forall u\in x\;\forall v\in u\;(v\in x).}
$$
This is a <bounded formula in set theory>. If $x\in M$, transitivity of $M$ ensures that every $u\in x$ and every $v\in u$ is already in $M$, and the same holds in $N$. Both structures therefore run through exactly the same bounded witnesses and use the same actual membership relation. They agree on $\varphi_1(x)$ for every $x\in M$, proving <set-theoretic absoluteness>.
Back to article page
