classes of ordinals and enumerating functions
A class of ordinals is just a subclass of the class (http://planetmath.org/Class) 𝐎𝐧 of all ordinals. For every class of ordinals M there is an enumerating function fM defined by transfinite recursion:
fM(α)=min{x∈M∣f(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 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 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 N⊆M such that |N|<κ, also .
We say is -unbounded if for any there is some such that .
We say a function is -continuous if is -closed and
A function is -normal if it is order preserving ( implies ) and continuous. In particular, the enumerating function of a -closed class is always -normal.
All these definitions can be easily extended to all ordinals: a class is closed (resp. unbounded) if it is -closed (unbounded) for all . A function is continuous (resp. normal) if it is -continuous (normal) for all .
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 |