 Solution

Work internally in the ground model and write $P=\mathrm{Levy}({\aleph }_0,\lambda )$. Conditions are finite partial functions with the coordinate-wise value bounds in the PDF, ordered by inclusion, so compatible conditions agree on their common domain and their union is a common strengthening.

Suppose there were $\lambda$ many pairwise incompatible conditions. Each fixed finite domain carries fewer than $\lambda$ possible functions: it uses finitely many ordinals below $\lambda$, and each coordinate has fewer than $\lambda$ choices. Regularity therefore lets us select $\lambda$ distinct domains. The finite-set form of the generalized delta-system lemma gives a subfamily of size $\lambda$ with common intersection $r$. There are fewer than $\lambda$ possible value assignments to the finite root $r$. Regularity gives a further subfamily of size $\lambda$ agreeing on the root. Any two of its conditions have a union in $P$, contradicting incompatibility. Thus $P$ has the $\lambda$-chain condition. Strong inaccessibility is more than is needed for this finite-support argument; regular uncountability suffices.

For each $0<\alpha <\lambda$ and $n<\omega$, requiring $(n,\alpha )$ to be in the domain is dense. For each $\xi <\alpha$, requiring $\xi$ to occur as a value at some $(n,\alpha )$ is also dense: choose a fresh $n$ and extend the condition. Hence the generic union defines, for each such $\alpha$, a surjection $g_{\alpha }:\omega \to \alpha$. Every ground ordinal below $\lambda$ is therefore countable in $M[G]$.

By the chain condition and the cardinal-preservation argument, $\lambda$ remains an uncountable cardinal in the extension; the same small-value argument preserves its regularity as well. All ordinals are unchanged. Since every smaller ordinal is countable and $\lambda$ is not,This is the finite Lévy collapse to omega-one. The dense-set construction proves the collapse below $\lambda$, while the chain condition is what prevents collapsing $\lambda$ itself.