# generalized continuum hypothesis

The *generalized continuum hypothesis* states that for any infinite^{} cardinal $\lambda $ there is no cardinal $\kappa $ such that $$.

An equivalent^{} condition is that ${\mathrm{\aleph}}_{\alpha +1}={2}^{{\mathrm{\aleph}}_{\alpha}}$ for every ordinal^{} $\alpha $.
Another equivalent condition is that ${\mathrm{\aleph}}_{\alpha}={\mathrm{\beth}}_{\alpha}$ for every ordinal $\alpha $.

Like the continuum hypothesis^{}, the generalized continuum hypothesis is known to be independent of the axioms of ZFC.

Title | generalized continuum hypothesis |

Canonical name | GeneralizedContinuumHypothesis |

Date of creation | 2013-03-22 12:05:31 |

Last modified on | 2013-03-22 12:05:31 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 15 |

Author | yark (2760) |

Entry type | Axiom |

Classification | msc 03E50 |

Synonym | generalised continuum hypothesis |

Synonym | GCH |

Related topic | AlephNumbers |

Related topic | BethNumbers |

Related topic | ContinuumHypothesis |

Related topic | Cardinality |

Related topic | CardinalExponentiationUnderGCH |

Related topic | ZermeloFraenkelAxioms |