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| 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 \emph{enumerating function} $f_M$ defined by transfinite recursion: |
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 \emph{enumerating function} $f_M$ defined by transfinite recursion: |
| $$f_M(\alpha)=\min\{x\in M\mid f(\beta)<x\text{ for all }\beta<\alpha\},$$ |
$$f_M(\alpha)=\min\{x\in M\mid f(\beta)<x\text{ for all }\beta<\alpha\},$$ |
| and we define the \emph{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 $M$. |
and we define the \emph{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 $M$. |
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| For an ordinal $\kappa$, we say $M$ is $\kappa$-\emph{closed} if for any $N\subseteq M$ such that $|N|<\kappa$, also $\sup N\in M$. |
For an ordinal $\kappa$, we say $M$ is $\kappa$-\emph{closed} if for any $N\subseteq M$ such that $|N|<\kappa$, also $\sup N\in M$. |
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| We say $M$ is \emph{$\kappa$-unbounded} if for any $\alpha<\kappa$ there is some $\beta\in M$ such that $\alpha<\beta$. |
We say $M$ is \emph{$\kappa$-unbounded} if for any $\alpha<\kappa$ there is some $\beta\in M$ such that $\alpha<\beta$. |
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We say a function $f\colon M\rightarrow\mathbf{On}$ is $\kappa$-\emph{continuous} if $M$ is $\kappa$-closed and
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We say a function $f:M\rightarrow\mathbf{On}$ is $\kappa$-\emph{continuous} if $M$ is $\kappa$-closed and
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| $$f(\sup N)=\sup \{f(\alpha)\mid \alpha\in N\}$$ |
$$f(\sup N)=\sup \{f(\alpha)\mid \alpha\in N\}$$ |
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| A function is \emph{$\kappa$-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. |
A function is \emph{$\kappa$-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. |
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| All these definitions can be easily extended to all ordinals: a class is \emph{closed} (resp. \emph{unbounded}) if it is $\kappa$-closed (unbounded) for all $\kappa$. A function is \emph{continuous} (resp. \emph{normal}) if it is $\kappa$-continuous (normal) for all $\kappa$. |
All these definitions can be easily extended to all ordinals: a class is \emph{closed} (resp. \emph{unbounded}) if it is $\kappa$-closed (unbounded) for all $\kappa$. A function is \emph{continuous} (resp. \emph{normal}) if it is $\kappa$-continuous (normal) for all $\kappa$. |
