classes of ordinals and enumerating functions (original) (raw)

For an ordinal κ, we say M is κ-closed if for any N⊆M such that |N|<κ, also sup⁡N∈M.

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

We say a function f:M→𝐎𝐧 is κ-continuous 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.