Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 24 / 6 / a

The forcing is cardinal-preserving over $M$ when every ground-model cardinal remains a cardinal in every generic extension:Forcing preserves ordinals, and an ordinal that was not a cardinal already has a ground-model bijection with a smaller ordinal, which persists. Thus this definition gives equality of the cardinal ordinals in the two models.

The forcing definability lemma states that for each formula $\phi (v_1,\dots ,v_n)$ there is a first-order formula ${\mathrm{Force}}_{\phi }$, depending on $\phi$ but not on the chosen names, such that for $p\in P$ and forcing names ${\tau }_i\in M$,The order relation of $\mathbb{P}$ is included in its set code. Thus each instance of the forcing relation is definable inside the ground model, allowing the uses of ground-model axiom schema of separation and replacement in the forcing arguments.

The forcing is $\kappa$-closed in $M$ when every descending sequence of conditions of length less than $\kappa$, belonging to $M$, has a lower bound in $M$:We use the convention that $q\le p$ means that $q$ is stronger. In particular, ${\omega }_1$-closed forcing is countably closed. Internal quantification over the sequences is essential when $M$ is externally countable.