A class of ordinals is just a subclass of the class $\mathbf{On}$ of all ordinals. For every class of ordinals $M$ there is an enumerating function $f_M$ defined by transfinite recursion: $$f_M(\alpha)=\min\{x\in M\mid f(\beta)<x\text{ for all }\beta<\alpha\},$$ and we define the order type of $M$ by $\operatorname{otype}(M)=\dom(f)$ . The possible values for this value are either $\mathbf{On}$ or some ordinal $\alpha$ . 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 $\alpha<\beta$ then $f_M(\alpha)<f_M(\beta)$ , so $f_M$ is an order isomorphism between $\operatorname{otype}(M)$ and $M$ .

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

We say $M$ 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 $M$ is $\kappa$ -closed and $$f(\sup N)=\sup \{f(\alpha)\mid \alpha\in N\}$$

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 $\kappa$ .