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# classes of ordinals and enumerating functions

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 $\operatorname{otype}(M)=\operatorname{dom}(f)$. 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 $\kappa$.

## Mathematics Subject Classification

03F15*no label found*03E10

*no label found*

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