# proof of Fodor’s lemma

If we let $f^{-1}:\kappa\rightarrow P(S)$ be the inverse of $f$ restricted to $S$ then Fodor’s lemma is equivalent to the claim that for any function such that $\alpha\in f(\kappa)\rightarrow\alpha>\kappa$ 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 $C_{\alpha}\cap f^{-1}(\alpha)=\emptyset$. Let $C=\Delta_{\alpha<\kappa}C_{\alpha}$. 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 $\beta<\alpha$, and so there can be no $\beta<\alpha$ such that $\alpha\in f^{-1}(\beta)$, so $f(\alpha)\geq\alpha$, a contradiction.

Title proof of Fodor’s lemma ProofOfFodorsLemma 2013-03-22 12:53:19 2013-03-22 12:53:19 Henry (455) Henry (455) 4 Henry (455) Proof msc 03E10