# fixed points of normal functions

If $f:M\to \mathrm{\mathbf{O}\mathbf{n}}$ is a function from any set of ordinals^{} to the class of ordinals then $\mathrm{Fix}(f)=\{x\in M\mid f(x)=x\}$ is the set of fixed points of $f$. ${f}^{\prime}$, the *derivative ^{}* of $f$, is the enumerating function of $\mathrm{Fix}(f)$.

If $f$ is $\kappa $-normal (http://planetmath.org/KappaNormal) then $\mathrm{Fix}(f)$ 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 $\ge \omega $, so ${f}^{\prime}(\alpha )=\omega +\alpha $.

Title | fixed points of normal functions |

Canonical name | FixedPointsOfNormalFunctions |

Date of creation | 2013-03-22 13:28:59 |

Last modified on | 2013-03-22 13:28:59 |

Owner | Henry (455) |

Last modified by | Henry (455) |

Numerical id | 6 |

Author | Henry (455) |

Entry type | Definition |

Classification | msc 03E10 |

Related topic | ProofOfPowerRule |

Related topic | LeibnizNotation |

Related topic | ProofOfProductRule |

Related topic | ProofOfSumRule |

Related topic | SumRule |

Related topic | DirectionalDerivative |

Related topic | NewtonsMethod |

Defines | derivative |