Finite-condition Lévy collapse ($\mathrm{Lv}(\kappa )$)

The finite-condition Lévy collapse $\mathrm{Lv}(\kappa )$ consists of finite partial functions $p$ with $\mathrm{dom}p\subseteq \kappa \times \omega$ and $p(\alpha ,n)<\alpha$, ordered by reverse inclusion. The generic union gives a surjection $\omega \to \alpha$ for every infinite $\alpha <\kappa$. If $\kappa$ is regular and uncountable, the Delta-system lemma at a regular uncountable cardinal proves the $\kappa$-chain condition, so the extension makes $\kappa ={\aleph }_1$.

With $\kappa$ an uncountable regular cardinal, finite partial functions on $\kappa \times \omega$ assigning a value below the first coordinate collapse every ground ordinal below $\kappa$ to countable size. The Delta-system lemma gives the $\kappa$ chain in a partial order condition; the possible-values lemma for chain-condition forcing then preserves the regularity of $\kappa$. Therefore $\kappa$ becomes the extension ${\omega }_1$. Strong inaccessibility is enough but is not needed for this identification.

For a regular uncountable $\kappa$, the finite-condition collapse has the $\kappa$ chain in a partial order condition, by a regular-cardinal Delta-system lemma and thinning to identical assignments on the finite root. Thus every maximal forcing antichain has ground-model size less than $\kappa$. There is no fixed size: for any nonzero cardinal $\mu <\kappa$, assigning all values below $\mu$ at the one coordinate $(\mu ,0)$ is a maximal forcing antichain of size $\mu$.