partial order with chain condition does not collapse cardinals

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.)

Title	partial order with chain condition does not collapse cardinals
Canonical name	PartialOrderWithChainConditionDoesNotCollapseCardinals
Date of creation	2013-03-22 12:53:40
Last modified on	2013-03-22 12:53:40
Owner	mathcam (2727)
Last modified by	mathcam (2727)
Numerical id	6
Author	mathcam (2727)
Entry type	Theorem
Classification	msc 03E35
Related topic	PartialOrder
Related topic	ChainCondition