{c}

A <cardinal-preserving forcing> over $M$ leaves every ground-model <cardinal number> a cardinal in every <generic extension>. In the present notation, for every $M$-generic filter $G$ and every ordinal $\kappa\in M$,
$$
\boxed{M\models\text{“}\kappa\text{ is a cardinal”}\quad\Longrightarrow\quad M[G]\models\text{“}\kappa\text{ is a cardinal”}.}
$$
Set <forcing> preserves ordinals, and an old bijection witnessing that an ordinal is not a cardinal remains available. Thus this condition also means that the two models have exactly the same cardinals. Cardinality here is computed inside the respective models, not from their external sizes.

Solution

Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 121 / 3 / iv / a / Solution

A forcing order preserves cardinals over a ground model if every ground-model <cardinal number> remains a cardinal in every <generic extension>. This assertion and any sufficient <chain condition for forcing> are evaluated inside the ground model. External countability of a <countable transitive model> does not make all of its forcing orders internally ccc.

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 Incoming links: Cardinal-preserving forcing