# ordinal number

An ordinal number^{} is a well ordered set $S$ such that, for every $x\in S$,

$$ |

(where $$ is the ordering relation on $S$).

It follows immediately from the definition that every ordinal is a transitive set. Also note that if $a,b\in S$ then we have $$ if and only if $a\in b$.

There is a theory of ordinal arithmetic which allows construction of various ordinals.
For example, all the numbers $0$, $1$, $2$, …have natural interpretations^{} as ordinals,
as does the set of natural numbers (including $0$),
which in this context is often denoted by $\omega $,
and is the first infinite^{} ordinal.

Title | ordinal number |
---|---|

Canonical name | OrdinalNumber |

Date of creation | 2013-03-22 12:07:55 |

Last modified on | 2013-03-22 12:07:55 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 8 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 03E10 |

Synonym | ordinal |

Related topic | VonNeumannOrdinal |