# transfinite induction

Suppose $\mathrm{\Phi}(\alpha )$ is a property defined for every ordinal^{} $\alpha $, the principle of *transfinite induction ^{}* states that in the case where for every $\alpha $, if the fact that $\mathrm{\Phi}(\beta )$ is true for every $$ implies that $\mathrm{\Phi}(\alpha )$ is true, then $\mathrm{\Phi}(\alpha )$ is true for every ordinal $\alpha $. Formally :

$$ |

The principle of transfinite induction is very similar to the principle of finite induction, except that it is stated in terms of the whole class of the ordinals.

Title | transfinite induction |
---|---|

Canonical name | TransfiniteInduction |

Date of creation | 2013-03-22 12:29:03 |

Last modified on | 2013-03-22 12:29:03 |

Owner | jihemme (316) |

Last modified by | jihemme (316) |

Numerical id | 10 |

Author | jihemme (316) |

Entry type | Theorem^{} |

Classification | msc 03B10 |

Synonym | principle of transfinite induction |

Related topic | PrincipleOfFiniteInduction |

Related topic | Induction^{} |

Related topic | TransfiniteRecursion |