# aleph numbers

The *aleph numbers* are infinite^{} cardinal numbers^{}
defined by transfinite recursion, as described below.
They are written ${\mathrm{\aleph}}_{\alpha}$, where $\mathrm{\aleph}$ is aleph,
the first letter of the Hebrew alphabet,
and $\alpha $ is an ordinal number^{}.
Sometimes we write ${\omega}_{\alpha}$ instead of ${\mathrm{\aleph}}_{\alpha}$,
usually to emphasise that it is an ordinal.

To start the transfinite recursion,
we define ${\mathrm{\aleph}}_{0}$ to be the first infinite ordinal.
This is the cardinality of countably infinite^{} sets, such as $\mathbb{N}$ and $\mathbb{Q}$.
For each ordinal $\alpha $,
the cardinal number ${\mathrm{\aleph}}_{\alpha +1}$ is defined to be
the least ordinal of cardinality greater than ${\mathrm{\aleph}}_{\alpha}$.
For each limit ordinal^{} $\delta $,
we define ${\mathrm{\aleph}}_{\delta}={\bigcup}_{\alpha \in \delta}{\mathrm{\aleph}}_{\alpha}$.

As a consequence of the Well-Ordering Principle (http://planetmath.org/ZermelosWellOrderingTheorem), every infinite set is equinumerous with an aleph number. Every infinite cardinal is therefore an aleph. More precisely, for every infinite cardinal $\kappa $ there is exactly one ordinal $\alpha $ such that $\kappa ={\mathrm{\aleph}}_{\alpha}$.

Title | aleph numbers |
---|---|

Canonical name | AlephNumbers |

Date of creation | 2013-03-22 14:15:39 |

Last modified on | 2013-03-22 14:15:39 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 6 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E10 |

Synonym | alephs |

Related topic | GeneralizedContinuumHypothesis |

Related topic | BethNumbers |