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

All cardinalities in this part are first computed in $M$. The forcing has size $\kappa$. A family of $\kappa$ finite domains has a $\kappa$-sized Delta-system, since $\kappa$ is regular. There are fewer than $\kappa$ possible value assignments on its finite root: each coordinate $(\alpha ,n)$ allows fewer than $\kappa$ values. Regularity lets us thin to two conditions, indeed $\kappa$ many, with identical root assignments. Their union is a condition, proving the $\kappa$-chain condition.

Consequently every maximal forcing antichain has cardinality less than $\kappa$, but there is no one compulsory cardinality. For any nonzero cardinal $\mu <\kappa$, the single-coordinate conditions assigning the values $\xi <\mu$ at $(\mu ,0)$ form an forcing antichain of size $\mu$. It is maximal: a condition already assigning that coordinate is compatible with its matching value, and a condition not assigning it is compatible with every allowed value. Thus every such size occurs, including singleton maximal forcing antichains. These are the maximal-antichain sizes in the finite Lévy collapse.

The paper writes $p\le q$ for $p\subseteq q$. We use this printed weaker-first convention: $q$ extends $p$ and is stronger. Compatibility and generic meeting arguments below always refer to common extensions, so do not accidentally reverse the convention.