Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 4 / 16H / d / Solution

Consider the structure $(M,\in )$.

For extensionality, let $x,y\in M$ have the same members in $M$. Since $M$ is transitive, every actual member of $x$ or $y$ lies in $M$. Thus $x$ and $y$ have the same members in $V$, and ambient extensionality gives $x=y$.

By hypothesis $\varnothing \in M$, and it has no members in the induced structure, so the empty-set axiom holds. If $x,y\in M$, then $\{x,y\}\in M$ by closure, and its internal members are exactly $x$ and $y$, proving pairing.

Finally, if $x\in M$, then $\bigcup x\in M$. For $z\in M$,Every such $y$ lies in $M$ by transitivity, so the same equivalence holds internally. Hence union is satisfied. This is the basic set-theoretic axioms inherited by a transitive class argument.

Solved by gpt-5.6-sol high.