Fodor's lemma

In mathematics, particularly in set theory, Fodor's lemma states the following:

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 ${\displaystyle \alpha \in S_{0}}$. In modern parlance, the nonstationary ideal is normal.

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