# 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 $$, $\lambda $ is also a cardinal in $\U0001d510[G]$, and if $\mathrm{cf}(\alpha )=\lambda $ in $\U0001d510$ then also $\mathrm{cf}(\alpha )=\lambda $ in $\U0001d510[G]$.

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 $\U0001d510$ 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 |
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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 |