= Solution
Let $F=\bigcup G$, the generic union for the <Finite-condition Lévy collapse>, and fix an infinite <cardinal number> $\lambda<\kappa$ in $M$. For every $n\in\omega$, the set of conditions defining a value at $(\lambda,n)$ is dense: extend a condition at that fresh coordinate with any value below $\lambda$. Hence $F_\lambda(n)=F(\lambda,n)$ is a total function $\omega\to\lambda$.
For every $\beta<\lambda$, the set of conditions assigning the value $\beta$ at some fresh coordinate $(\lambda,n)$ is also dense. Genericity therefore makes $F_\lambda$ surjective. Thus
$$
\boxed{F_\lambda:\omega\twoheadrightarrow\lambda,}
$$
so $\lambda$ is a <countable set> in $M[G]$. Finite cardinals are already countable.
Back to article page
