= Finite Lévy collapse to omega-one
{title2=$\kappa=\omega_1^{M[G]}$}
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.
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