= Cardinal preservation by closed forcing
{title2=$M[G]\models\text{“}\lambda\text{ is a cardinal”}$}
A $\lambda$-<closed forcing> order preserves a ground-model <cardinal number> $\lambda$, even when it is singular. A proposed surjection from an ordinal below $\lambda$ to $\lambda$ would be an old function because <closed forcing adds no short ordinal sequences>, contradicting old cardinalhood. This concerns $\lambda$ and smaller cardinals; closure alone does not supply preservation of all larger cardinals.
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