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For every tuple of parameters $\bar{a}\in M^n$, an upward absolute formula transfers truth from the smaller structure to the larger one:These definitions of directional set-theoretic absoluteness do not themselves require the structures to be transitive.

A formula has set-theoretic absoluteness between these structures when both directions hold for every $\bar{a}\in M^n$:Equivalently, it is both a downward absolute formula and an upward absolute formula between them.

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.

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.

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.

Let $\alpha =({\omega }_1{)}^M$ and takeThis formula is upward absolute between transitive models: if the smaller model contains such an $f$, the assumed absoluteness of “function”, domain, range, and $\omega$ shows that the same witness works in the larger model.

The generic union $g=\bigcup G$ is a total map $\omega \to \alpha$, because the conditions deciding each input form a dense set. For every $\beta <\alpha$, the conditions putting $\beta$ somewhere in the range are also dense, so $g$ is surjective. Thus $M[G]\models {\phi }_1(\alpha )$. But $M/\models {\phi }_1(\alpha )$ because $M$ regards $\alpha$ as its first uncountable ordinal. Hence ${\phi }_1$ is not downward absolute between $M$ and $M[G]$.