= Solution
{c}
Let $C(x)$ be the formula expressing <cardinal number> status:
$$
\boxed{\varphi_2(x)=C(x):\quad x\text{ is an ordinal}\ \land\ \forall\alpha\in x\;\neg\exists f\;(f\text{ is a bijection }\alpha\to x).}
$$
Ordinalhood and the assertion that a given set is a <bijection> between fixed sets are each expressible by a <bounded formula in set theory>. If $M$ regarded $x$ as a non-cardinal ordinal, a witnessing smaller ordinal and bijection would remain present and valid in $N$. Ordinalhood itself is absolute between the <transitive models>. Thus $N\models C(x)$ implies $M\models C(x)$: this is <downward absoluteness of cardinalhood>.
For failure in the other direction, put $\beta=(\omega_1)^M$. This ordinal belongs to $M$, and $M\models C(\beta)$ with $\beta>\omega$. Externally $\beta$ is countable, since $\beta\subseteq M$ and $M$ is countable. By the construction of $N$, it contains a countability witness for $\beta$, and hence a bijection $\omega\to\beta$. Therefore $N\models\neg C(\beta)$. This formula is downward absolute and fails to be upward absolute.
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