Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 19 / 6 / i / a

A forcing collapses a cardinal $\kappa$ over $M$ if it forces that the ground ordinal $\kappa$ is no longer a cardinal: in the extension there is a bijection from some smaller ordinal onto $\kappa$. Equivalently there is a surjection onto $\kappa$ from an ordinal whose ground cardinality is smaller. Forcing preserves the ordinal itself; collapse changes cardinality, not the identity of the ordinal. If the assertion is about one particular generic extension, the witnessing collapse is interpreted there; a forcing-wide assertion means it is forced by the weakest condition.