# limit cardinal

A limit cardinal is a cardinal $\kappa $ such that $$ for every cardinal $$. Here ${\lambda}^{+}$ denotes the cardinal successor of $\lambda $. If $$ for every cardinal $$, then $\kappa $ is called a strong limit cardinal.

Every strong limit cardinal is a limit cardinal, because ${\lambda}^{+}\le {2}^{\lambda}$ holds for every cardinal $\lambda $.
Under GCH, every limit cardinal is a strong limit cardinal because in this case ${\lambda}^{+}={2}^{\lambda}$ for every infinite^{} cardinal $\lambda $.

The three smallest limit cardinals are $0$, ${\mathrm{\aleph}}_{0}$ and ${\mathrm{\aleph}}_{\omega}$.
Note that some authors do not count $0$, or sometimes even ${\mathrm{\aleph}}_{0}$, as a limit cardinal.
An infinite cardinal ${\mathrm{\aleph}}_{\alpha}$ is a limit cardinal
if and only if $\alpha $ is either $0$ or a limit ordinal^{}.

Title | limit cardinal |
---|---|

Canonical name | LimitCardinal |

Date of creation | 2013-03-22 14:04:40 |

Last modified on | 2013-03-22 14:04:40 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 15 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E10 |

Related topic | SuccessorCardinal |

Defines | strong limit cardinal |