Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 121 / 3 / i

For a condition $p\in \mathbb{P}$, a set $D\subseteq \mathbb{P}$ is dense below a forcing condition $p$ whenThus every extension of $p$ can be strengthened into $D$.

A set $G\subseteq \mathbb{P}$ is a filter when it is upward closed toward weaker conditions and downward directed: if $p\in G$ and $p\le q$, then $q\in G$, while any $p,q\in G$ have some $r\in G$ with $r\le p,q$. It is a generic filter over $M$ when

The forcing order $\mathbb{P}$ has the chain condition for forcing at $\kappa$ when every antichain in a forcing order has cardinality below $\kappa$:When this assertion is evaluated inside $M$, both the quantified subsets and their cardinalities are those of $M$.