Past exam of the mathematics course of the University of Cambridge / 2013 / iii / Paper 19 / 1 / ii / b / Solution

The printed inclusion-and-elementarity assertion is false if transitivity is required of the same submodel. Take the theorem of ZFC that combines the axiom of infinity and the Axiom of power set. Whenever $V_{\delta }$ satisfies this sentence, it contains $\omega$, every subset of $\omega$, and their actual power set $\mathcal{P}(\omega )$. A submodel $M\prec V_{\delta }$ contains $\omega$ and $\mathcal{P}(\omega )$, since these are uniquely definable in $V_{\delta }$. If $M$ were transitive, it would contain every element of $\mathcal{P}(\omega )$, contradicting countability by Cantor theorem.

The corrected conclusion uses an elementary embedding rather than elementary inclusion. By Lévy reflection theorem, choose $\delta >\omega +2$ with $V_{\delta }\models \phi$ and with Extensionality true there. The Downward Lowenheim-Skolem theorem says that an infinite structure in a countable language has a countable elementary substructure. Apply it to obtain a countable $N\prec (V_{\delta },\in )$. The membership relation on $N$ is externally well-founded, and elementarity makes it extensional. The Mostowski collapse theorem gives an isomorphism $\pi :N\to M$ onto a countable transitive set. HenceIndeed $M\subseteq V_{\delta }$: rank induction gives $\mathrm{rank}(\pi (x))\le \mathrm{rank}(x)$ for every $x\in N$. The crucial correction is that $j$, rather than the inclusion of $M$, is elementary. For a formula with free variables, apply this argument to its universal closure.