partial order with chain condition does not collapse cardinalsPartialOrderWithChainConditionDoesNotCollapseCardinals2002-07-30 21:55:542004-03-27 08:18:05Theorem% this is the default PlanetMath preamble. as your knowledge
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If $P$ is a partial order which satisfies the $\kappa$ chain condition and $G$ is a generic subset of $P$ then for any $\kappa<\lambda\in\mathfrak{M}$, $\lambda$ is also a cardinal in $\mathfrak{M}[G]$, and if $\operatorname{cf}(\alpha)=\lambda$ in $\mathfrak{M}$ then also $\operatorname{cf}(\alpha)=\lambda$ in $\mathfrak{M}[G]$.
This theorem is the simplest way to control a notion of forcing, since it means that a notion of forcing does not have an effect above a certain point. Given that any $P$ satisfies the $|P|^+$ chain condition, this means that most forcings leaves all of $\mathfrak{M}$ above a certain point alone. (Although it is possible to get around this limit by forcing with a proper class.)