# proof of principle of transfinite induction

To prove the transfinite induction^{} theorem^{}, we note that the class of ordinals^{} is well-ordered by $\in $. So suppose for some $\mathrm{\Phi}$, there are ordinals $\alpha $ such that $\mathrm{\Phi}(\alpha )$ is not true. Suppose further that $\mathrm{\Phi}$ satisfies the hypothesis^{}, i.e.
$$. We will reach a contradiction^{}.

The class $C=\{\alpha :\mathrm{\neg}\mathrm{\Phi}(\alpha )\}$ is not empty. Note that it may be a proper class^{}, but this is not important. Let $\gamma =\mathrm{min}(C)$ be the $\in $-minimal element of $C$. Then by assumption^{}, for every $$, $\mathrm{\Phi}(\lambda )$ is true. Thus, by hypothesis, $\mathrm{\Phi}(\gamma )$ is true, contradiction.

Title | proof of principle of transfinite induction |
---|---|

Canonical name | ProofOfPrincipleOfTransfiniteInduction |

Date of creation | 2013-03-22 12:29:06 |

Last modified on | 2013-03-22 12:29:06 |

Owner | jihemme (316) |

Last modified by | jihemme (316) |

Numerical id | 11 |

Author | jihemme (316) |

Entry type | Proof |

Classification | msc 03B10 |