# proof of Fodor’s lemma

If we let ${f}^{-1}:\kappa \to P(S)$ be the inverse^{} of $f$ restricted to $S$ then Fodor’s lemma is equivalent^{} to the claim that for any function such that $\alpha \in f(\kappa )\to \alpha >\kappa $ there is some $\alpha \in S$ such that ${f}^{-1}(\alpha )$ is stationary.

Then if Fodor’s lemma is false, for every $\alpha \in S$ there is some club set ${C}_{\alpha}$ such that ${C}_{\alpha}\cap {f}^{-1}(\alpha )=\mathrm{\varnothing}$. Let $$. The club sets are closed under diagonal intersection, so $C$ is also club and therefore there is some $\alpha \in S\cap C$. Then $\alpha \in {C}_{\beta}$ for each $$, and so there can be no $$ such that $\alpha \in {f}^{-1}(\beta )$, so $f(\alpha )\ge \alpha $, a contradiction^{}.

Title | proof of Fodor’s lemma |
---|---|

Canonical name | ProofOfFodorsLemma |

Date of creation | 2013-03-22 12:53:19 |

Last modified on | 2013-03-22 12:53:19 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 4 |

Author | Henry (455) |

Entry type | Proof |

Classification | msc 03E10 |