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.)

单词	PartialOrderWithChainConditionDoesNotCollapseCardinals
释义	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.)
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