# classes of ordinals and enumerating functions

A class of ordinals is just a subclass of the class (http://planetmath.org/Class) $\mathbf{On}$ of all ordinals. For every class of ordinals $M$ there is an enumerating function $f_{M}$ defined by transfinite recursion:

 $f_{M}(\alpha)=\min\{x\in M\mid f(\beta)

and we define the order type of $M$ by $\operatorname{otype}(M)=\operatorname{dom}(f)$. The possible values for this value are either $\mathbf{On}$ 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 $\alpha<\beta$ then $f_{M}(\alpha), so $f_{M}$ is an order isomorphism between $\operatorname{otype}(M)$ and $M$.

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

We say $M$ is $\kappa$-unbounded if for any $\alpha<\kappa$ there is some $\beta\in M$ such that $\alpha<\beta$.

We say a function $f\colon M\rightarrow\mathbf{On}$ is $\kappa$-continuous if $M$ is $\kappa$-closed and

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

A function is $\kappa$-normal if it is order preserving ($\alpha<\beta$ implies $f(\alpha)) 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