# von Neumann ordinal

The is a method of defining ordinals^{} in set theory^{}.

The von Neumann ordinal $\alpha $ is defined to be the well-ordered set containing the von Neumann ordinals which precede $\alpha $. The set of finite von Neumann ordinals is known as the von Neumann integers. Every well-ordered set is isomorphic to a von Neumann ordinal.

They can be constructed by transfinite recursion as follows:

If an ordinal is the successor of another ordinal, it is an *successor ordinal*. If an ordinal is neither $0$ nor a successor ordinal then it is a *limit ordinal*. The first limit ordinal is named $\omega $.

The class of ordinals is denoted $\mathrm{\mathbf{O}\mathbf{n}}$.

The von Neumann ordinals have the convenient property that if $$ then $a\in b$ and $a\subset b$.

Title | von Neumann ordinal |

Canonical name | VonNeumannOrdinal |

Date of creation | 2013-03-22 12:32:37 |

Last modified on | 2013-03-22 12:32:37 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 11 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03E10 |

Synonym | ordinal |

Related topic | VonNeumannInteger |

Related topic | ZermeloFraenkelAxioms |

Related topic | OrdinalNumber |

Defines | successor ordinal |

Defines | limit ordinal |

Defines | successor |