# beth numbers

The *beth numbers* are infinite^{} cardinal numbers^{}
defined in a similar manner to the aleph numbers, as described below.
They are written ${\mathrm{\beth}}_{\alpha}$, where $\mathrm{\beth}$ is beth,
the second letter of the Hebrew alphabet,
and $\alpha $ is an ordinal number^{}.

We define ${\mathrm{\beth}}_{0}$ to be the first infinite cardinal (that is, ${\mathrm{\aleph}}_{0}$).
For each ordinal $\alpha $,
we define ${\mathrm{\beth}}_{\alpha +1}={2}^{{\mathrm{\beth}}_{\alpha}}$.
For each limit ordinal^{} $\delta $,
we define ${\mathrm{\beth}}_{\delta}={\bigcup}_{\alpha \in \delta}{\mathrm{\beth}}_{\alpha}$.

Note that ${\mathrm{\beth}}_{1}$ is the cardinality of the continuum^{}.

For any ordinal $\alpha $ the inequality ${\mathrm{\aleph}}_{\alpha}\u2a7d{\mathrm{\beth}}_{\alpha}$ holds.
The Generalized Continuum Hypothesis is equivalent^{} to the assertion that
${\mathrm{\aleph}}_{\alpha}={\mathrm{\beth}}_{\alpha}$ for every ordinal $\alpha $.

For every limit ordinal $\delta $, the cardinal ${\mathrm{\beth}}_{\delta}$ is a strong limit cardinal. Every uncountable strong limit cardinal arises in this way.

Title | beth numbers |
---|---|

Canonical name | BethNumbers |

Date of creation | 2013-03-22 14:17:09 |

Last modified on | 2013-03-22 14:17:09 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 9 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E10 |

Related topic | AlephNumbers |

Related topic | GeneralizedContinuumHypothesis |