\b[U.] A witness in the smaller model that $x$ is a <countable set> remains a function in the larger model, so countability is described by an <upward absolute formula>. It is not downward absolute: the larger model may have a new enumeration of $x$ that is absent from the smaller model, as explained by <upward absoluteness of countability>.

Solution

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

If a smaller transitive model has a function witnessing that $x$ is a <countable set>, the same witness exists in every larger transitive model. Downward absoluteness can fail because a larger model may contain a new enumeration of $x$.

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 Incoming links: Upward absoluteness of countability