= Absoluteness of cardinalhood in limit ranks
For <limit ordinals> $\alpha\leq\beta$, the structures $V_\alpha$ and $V_\beta$ agree on whether any shared <set> is a <cardinal number>. They agree on being an <ordinal>. If $\delta<\alpha$ is not a <cardinal number>, a <bijection> from some $\gamma<\delta$ onto $\delta$ has <rank of a set> at most $\delta+3$, hence belongs to $V_\alpha$. Its defining properties are <bounded formulas in set theory>. Both structures therefore see the same failure witness. This full agreement is stronger than <downward absoluteness of cardinalhood> between arbitrary <transitive models>.
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