Cardinal effects of collapse and Lévy-collapse forcing

A descending sequence of fewer than $\kappa$ collapse conditions has union of domain size below $\kappa$, so $\mathrm{Col}(\kappa ,\lambda )$ is $\kappa$-closed. For each $\xi <\kappa$ and $\beta <\lambda$, the sets requiring $\xi$ in the domain and $\beta$ in the range are dense (using a fresh coordinate for the latter). Hence the generic union is a total surjection $\kappa \twoheadrightarrow \lambda$.

Given $\theta$ many Lévy conditions, F3 thins their finite domains to a delta system with a fixed finite root. At each root coordinate $(\alpha ,n)$ there are only $\mathrm{\mid }\alpha \mathrm{\mid }<\theta$ possible values; regularity and finiteness of the root therefore leave fewer than $\theta$ possible root assignments. Thin the $\theta$ conditions until their root restrictions agree. Two remaining conditions then have compatible union, so $\mathrm{Lv}(\theta )$ is $\theta$-cc.

For every $0<\alpha <\theta$, the union of the generic restrictions to $\{\alpha \}\times \omega$ is total by coordinate dense sets and hits every $\beta <\alpha$ by range dense sets. It is a surjection $\omega \twoheadrightarrow \alpha$. step 1.2 and F2 preserve the cardinal and regularity of $\theta$, while all smaller infinite ordinals become countable; hence the extension identifies $\theta$ with ${\mathrm{\aleph }}_1$.