classes of ordinals and enumerating functions
and we define the order type of by . The possible values for this value are either or some ordinal . 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 then , so is an order isomorphism between and .
For an ordinal , we say is -closed if for any such that , 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|
|Date of creation||2013-03-22 13:28:55|
|Last modified on||2013-03-22 13:28:55|
|Last modified by||mathcam (2727)|