classes of ordinals and enumerating functions

A class of ordinalsMathworldPlanetmathPlanetmath is just a subclass of the class ( 𝐎𝐧 of all ordinals. For every class of ordinals M there is an enumerating function fM defined by transfinite recursion:

fM(α)=min{xMf(β)<x for all β<α},

and we define the order type of M by otype(M)=dom(f). The possible values for this value are either 𝐎𝐧 or some ordinal α. The above functionMathworldPlanetmath 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 α<β then fM(α)<fM(β), so fM is an order isomorphism between otype(M) and M.

For an ordinal κ, we say M is κ-closed if for any NM such that |N|<κ, also supNM.

We say M is κ-unboundedPlanetmathPlanetmath if for any α<κ there is some βM such that α<β.

We say a function f:M𝐎𝐧 is κ-continuousMathworldPlanetmath if M is κ-closed and


A function is κ-normal if it is order preserving (α<β implies f(α)<f(β)) and continuous. In particular, the enumerating function of a κ-closed class is always κ-normal.

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

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