# classes of ordinals and enumerating functions

A *class of ordinals ^{}* is just a subclass of the class (http://planetmath.org/Class) $\mathrm{\mathbf{O}\mathbf{n}}$ of all ordinals. For every class of ordinals $M$ there is an

*enumerating function*${f}_{M}$ defined by transfinite recursion:

$$ |

and we define the *order type* of $M$ by $\mathrm{otype}(M)=\mathrm{dom}(f)$. The possible values for this value are either $\mathrm{\mathbf{O}\mathbf{n}}$ or some ordinal $\alpha $. The above function^{} simply lists the elements of $M$ in order. Note that it is not necessarily defined for all ordinals, although it is defined for a segment of the ordinals. If $$ then $$, so ${f}_{M}$ is an order isomorphism between $\mathrm{otype}(M)$ and $M$.

For an ordinal $\kappa $, we say $M$ is $\kappa $-*closed* if for any $N\subseteq M$ such that $$, also $supN\in M$.

We say $M$ is *$\kappa $-unbounded ^{}* if for any $$ there is some $\beta \in M$ such that $$.

We say a function $f:M\to \mathrm{\mathbf{O}\mathbf{n}}$ is $\kappa $-*continuous ^{}* if $M$ is $\kappa $-closed and

$$f(supN)=sup\{f(\alpha )\mid \alpha \in N\}$$ |

A function is *$\kappa $-normal* if it is order preserving ($$ implies $$) and continuous. In particular, the enumerating function of a $\kappa $-closed class is always $\kappa $-normal.

All these definitions can be easily extended to all ordinals: a class is *closed* (resp. *unbounded*) if it is $\kappa $-closed (unbounded) for all $\kappa $. A function is *continuous* (resp. *normal*) if it is $\kappa $-continuous (normal) for all $\kappa $.

Title | classes of ordinals and enumerating functions |

Canonical name | ClassesOfOrdinalsAndEnumeratingFunctions |

Date of creation | 2013-03-22 13:28:55 |

Last modified on | 2013-03-22 13:28:55 |

Owner | mathcam (2727) |

Last modified by | mathcam (2727) |

Numerical id | 14 |

Author | mathcam (2727) |

Entry type | Definition |

Classification | msc 03F15 |

Classification | msc 03E10 |

Defines | order type |

Defines | enumerating function |

Defines | closed |

Defines | kappa-closed |

Defines | continuous |

Defines | kappa-continuous |

Defines | continuous function |

Defines | kappa-continuous function |

Defines | closed class |

Defines | kappa-closed class |

Defines | normal function |

Defines | kappa-normal function |

Defines | normal |

Defines | kappa-normal |

Defines | unbounded |

Defines | unbounded clas |