\b[N.] A larger model may collapse a singular <cardinal number>, destroying cardinalhood, so the assertion is not described by an <upward absolute formula>. It may instead add a short <cofinal function> to a <regular cardinal> in the smaller model, so it is not described by a <downward absolute formula>. This is the <nonabsoluteness of singular cardinalhood>.

Solution

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

The statement that $\kappa$ is a <cardinal number> with $\operatorname{cf}(\kappa)<\kappa$ is generally neither upward nor downward absolute. Upward absoluteness can fail when a larger model collapses the cardinal; downward absoluteness can fail when a larger model adds a short cofinal sequence to a cardinal that the smaller model regards as regular.

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 Incoming links: Nonabsoluteness of singular cardinalhood