PlanetMath: classes of ordinals and enumerating functions

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classes of ordinals and enumerating functions

(Definition)

A class of ordinals is just a subclass of the class $\mathbf{On}$ of all ordinals. For every class of ordinals there is an enumerating function defined by transfinite recursion:

$\displaystyle f_M(\alpha)=\min\{x\in M\mid f(\beta)<x$ for all $\displaystyle \beta<\alpha\},$

and we define the order type of

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

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

is an order isomorphism between $\operatorname{otype}(M)$ and

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

We say 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 is $\kappa$ -closed and

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

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 $\kappa$ .

"classes of ordinals and enumerating functions" is owned by mathcam. [ full author list (3) | owner history (2) ]

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Also defines:

order type, enumerating function, closed, kappa-closed, continuous, kappa-continuous, continuous function, kappa-continuous function, closed class, kappa-closed class, normal function, kappa-normal function, normal, kappa-normal, unbounded, unbounded class, kappa-unbounded, kappa-unbounded class, class of ordinals

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Cross-references: definitions, implies, isomorphism, segment, function, transfinite recursion, ordinals, subclass
There are 27 references to this entry.

This is version 11 of classes of ordinals and enumerating functions, born on 2003-02-23, modified 2006-10-28.
Object id is 4053, canonical name is ClassesOfOrdinalsAndEnumeratingFunctions.
Accessed 28698 times total.

Classification:

AMS MSC:	03E10 (Mathematical logic and foundations :: Set theory :: Ordinal and cardinal numbers)
	03F15 (Mathematical logic and foundations :: Proof theory and constructive mathematics :: Recursive ordinals and ordinal notations)

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