\b[D.] <Downward absoluteness of cardinalhood> holds because, if the larger <transitive model> sees no <bijection> with a smaller <ordinal>, neither can the smaller model, whose functions form only a subset of those in the larger model. Cardinalhood is not described by an <upward absolute formula>, because the larger model can contain a new bijection collapsing an ordinal that the smaller model regards as a <cardinal number>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 121 / 1 / i / b / Solution

If a larger transitive model regards an ordinal $\kappa$ as a <cardinal number>, then a smaller transitive model does too: any bijection in the smaller model witnessing otherwise would remain in the larger one. The converse can fail when the larger model contains a new bijection between $\kappa$ and a smaller ordinal.

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