Fodor's lemma

Remarks

The proof is by contradiction. If each fibre $f^{-1}(\gamma)$ is nonstationary, pick a club set $C_\gamma$ disjoint from it. The diagonal intersection $D = \set{\alpha < \kappa : \alpha \in \bigcap_{\gamma < \alpha} C_\gamma}$ is club, so it meets $S$ in some ordinal $\alpha$. Then $\gamma = f(\alpha) < \alpha$ gives $\alpha \in C_\gamma$, which contradicts $f(\alpha) = \gamma$. Example: on the limit ordinals below $\omega_1$, any function with $f(\alpha) < \alpha$ is constant on a stationary set. Stationarity is necessary: on the successor ordinals, $f(\beta + 1) = \beta$ is regressive and injective, so all its fibres have one element. The lemma is also called the pressing-down lemma.

Going deeper

For a student

Fodor's lemma is the basic tool for working with stationary sets. It is the infinite analogue of the pigeonhole principle for regressive functions, and it relies on closure of the club filter under diagonal intersections. It is used in Solovay's splitting theorem for stationary sets and in Silver's theorem on singular cardinals.