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)<x\text{ for all }\beta<\alpha\},

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 $f_{M}(\alpha)<f_{M}(\beta)$ , 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 $f(\alpha)<f(\beta)$ ) 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