= Cardinal preservation by chain-condition forcing
If $\kappa$ is a <regular cardinal> and $\mathbb P$ has the $\kappa$-chain condition, then forcing with $\mathbb P$ preserves every <cardinal number> and <cofinality> at least $\kappa$. Indeed, a proposed surjection $f:\mu\to\lambda$ with $\mu<\lambda$ has its range contained in $\bigcup_{\xi<\mu}B_\xi$ from the <possible-values lemma for chain-condition forcing>. The regularity of $\kappa$ and <infinite cardinal arithmetic> make this union have cardinality below the ground-model cardinal $\lambda$, a contradiction.
Back to article page
