Fodor's lemma

Fodor's lemma—If ${\displaystyle \kappa }$ is a regular, uncountable cardinal, ${\displaystyle S}$ is a stationary subset of ${\displaystyle \kappa }$, and ${\displaystyle f:S\rightarrow \kappa }$ is regressive (that is, ${\displaystyle f(\alpha )<\alpha }$ for any ${\displaystyle \alpha \in S}$, ${\displaystyle \alpha \neq 0}$) then there is some ${\displaystyle \gamma }$ and some stationary ${\displaystyle S_{0}\subseteq S}$ such that ${\displaystyle f(\alpha )=\gamma }$ for any ${\displaystyle \alpha \in S_{0}}$.

Fodor's lemma for trees—For every non-special tree ${\displaystyle T}$ and regressive mapping ${\displaystyle f:T\rightarrow T}$ (that is, ${\displaystyle f(t)<t}$, with respect to the order on ${\displaystyle T}$, for every ${\displaystyle t\in T}$, ${\displaystyle t\neq 0}$), there is a non-special subtree ${\displaystyle S\subseteq T}$ on which ${\displaystyle f}$ is constant.