fixed points of normal functionsFixedPointsOfNormalFunctions2003-02-23 19:55:242004-10-10 10:27:14Definitionderivative% this is the default PlanetMath preamble. as your knowledge
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%\PMlinkescapeword{theory}If $f\colon M\rightarrow\mathbf{On}$ is a function from any set of ordinals to the class of ordinals then $\operatorname{Fix}(f)=\{x\in M\mid f(x)=x\}$ is the set of fixed points of $f$. $f^\prime$, the \emph{derivative} of $f$, is the enumerating function of $\operatorname{Fix}(f)$.
If $f$ is \PMlinkname{$\kappa$-normal}{KappaNormal} then $\operatorname{Fix}(f)$ is $\kappa$-closed and $\kappa$-normal, and therefore $f^\prime$ is also $\kappa$-normal.
For example, the function which takes an ordinal $\alpha$ to the ordinal $1+\alpha$ has a fixed point at every ordinal $\geq\omega$, so $f^\prime(\alpha)=\omega+\alpha$.