Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 5 / i

A forcing name is built by well-founded recursion: every member of a $\mathbb{P}$-name $\tau$ is a pair $(\sigma ,p)$ with $p\in \mathbb{P}$ and $\sigma$ itself a $\mathbb{P}$-name of lower forcing name rank. In the ground model all such forcing names form the recursively defined class $V^{\mathbb{P}}$. Its evaluation by a generic filter isThe recursion is on forcing name rank, not on the forcing order. Forcing names need not have a unique evaluation across different generics, and many different forcing names may have the same evaluation.

The forcing truth lemma says that for a formula $\phi$ and ground-model forcing names ${\tau }_1,\dots ,{\tau }_n$,The forcing relation is the ground-model relation furnished by the forcing theorem. The witnessing condition must belong to the particular generic filter, not merely to the forcing order.

A forcing preserves cofinalities over $M$ if every generic extension hasThe two values are compared as ordinals, using preservation of ordinals by forcing. Equivalently the forcing asserts that no ground ordinal acquires a smaller cofinality. The assertion concerns all ordinal cofinalities, not just that one specified cardinal remains uncountable.