= Solution
{c}
For the model $M$ constructed in part (a), the collapsing map $X\to M$ is a <bijection>. The <Downward Lowenheim-Skolem theorem> supplied countable $X$, so $M$ is externally countable. It is infinite because it satisfies the <axiom of infinity>. Consequently
$$
\boxed{|M|=\aleph_0.}
$$
This is external <cardinality>. A <countable transitive model> still regards many of its own sets and ordinals as uncountable, because external enumerations need not belong to it.
Back to article page
