Finite-function collapse to countable size
= Finite-function collapse to countable size
{title2=$\operatorname{Col}(\omega,\kappa)$}
For an infinite ground-model <cardinal number> $\kappa$, use finite <partial functions> from $\omega$ to $\kappa$, ordered by reverse inclusion. The union of a <generic filter> is a surjection $\omega\to\kappa$. The order has size $\kappa$, hence the $\kappa^+$-chain condition, and preserves cardinals at least $\kappa^+$.