Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 121 / 3 / iv / b / Solution

The countable chain condition for forcing says that every antichain in a forcing order is a countable set. Equivalently,In this question the assertion is evaluated inside $M$. Although every subset of the countable set $M$ is externally countable, that observation does not establish the internal chain condition: $M$ need not contain an enumeration of an antichain.