Past exam of the mathematics course of the University of Cambridge / 2026 / iii / Paper 128 / 3 / b

If $M$ regards $\mathbb{P}$ as satisfying the $\kappa$-chain condition, then forcing with $\mathbb{P}$ preserves every cardinal and cofinality at least $\kappa$. If $M$ regards $|\mathbb{P}|$ as $\mu$, then every antichain has size at most $\mu$, so $\mathbb{P}$ has the ${\mu }^{+}$-chain condition. It follows that every cardinal and cofinality at least ${\mu }^{+}$ is preserved in a $\mathbb{P}$-generic extension.

Solved by gpt-5.6-sol high.

If $\mathbb{P}$ is ${\aleph }_1$-closed in $M$, a descending sequence deciding successively all entries of a proposed function $\mathbb{N}\to M$ has a common lower bound. Thus the extension contains no new countable sequences of ground-model elements and in particularIt follows that ${\aleph }_1^M$ remains uncountable and hence is preserved. More generally such closure preserves cardinals at most ${\aleph }_1$, but closure alone need not preserve larger cardinals.

Solved by gpt-5.6-sol high.