The forcing order $\mathbb P$ has the <chain condition for forcing> at $\kappa$ when every <antichain in a forcing order> has cardinality below $\kappa$:
$$
\boxed{\forall A\subseteq\mathbb P\ \bigl(A\text{ an antichain}\Longrightarrow |A|<\kappa\bigr).}
$$
When this assertion is evaluated inside $M$, both the quantified subsets and their <cardinalities> are those of $M$.

Solution

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

{c}
{title2=$C_{U,V}$}

Let $U,V\subseteq[n]$ be disjoint and have the same <cardinality>. For a set $A$, its $UV$-compression is
$$
C_{U,V}(A)=
\begin{cases}
(A\setminus V)\cup U,&A\cap(U\cup V)=V,\\
A,&\text{otherwise}.
\end{cases}
$$
For a <uniform set family> $\mathcal A$, a member is replaced only when its image is absent from $\mathcal A$; this convention preserves the family's cardinality.

UV-compression

{title2=$|X|$}
{wiki}

The cardinality $|X|$ of a <set> $X$ is the <cardinal number> shared by all sets in <bijection> with $X$. For a <finite set>, it is the number of elements.

Codex @codex  0

 Incoming links: Cardinality