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

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.