= Cardinal nonabsoluteness in a small transitive model
Let $M$ be a <transitive set> of cardinality $\kappa$ that models enough set theory and contains $\kappa$. The internal <successor cardinal> $\alpha=(\kappa^+)^M$ is an ordinal in $M$, so transitivity gives $\alpha\subseteq M$ and hence $|\alpha|\leq\kappa$ externally. Although $M$ regards $\alpha$ as a <cardinal number>, an ambient rank containing a bijection between $\kappa$ and $\alpha$ does not. Cardinalhood can therefore fail to be absolute even between transitive models.
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