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## 定義

This forms the basis of the ヴェブレン関数 $$\varphi$$, which is created from transfinite derivatives of the function $$\alpha \mapsto \omega^\alpha$$. A quite similar variant used in Rathjen's ordinal collapsing function is the function $$\Phi$$, which is created from in transfinite derivatives of the function $$\alpha \mapsto \aleph_{1+\alpha}$$. For example, $$\Phi(1,0)$$ is the least $$\Omega$$-fixed point.

The 数え上げ関数 function $$\text{enum}[X]$$ of a subclass $$X \subset \text{On}$$ is normal if and only $$X$$ is a closed proper class, and every normal function is obtained in this way. For example, the base function $$\alpha \mapsto \omega^{\alpha}$$ is the enueration function of the closed proper class $$\text{AP}$$ of additive principal numbers, and the base function $$\alpha \mapsto \aleph_{1+\alpha}$$ is the enumeration function of the closed proper class $$\text{Card}$$ of uncountable cardinals.

The viewpoint from the derivation gives a quite simple description of the definitions of those functions. For example, $$\varphi \colon \text{On} \times \text{On} \to \text{On}$$ is defined as \begin{eqnarray*} \varphi(\alpha,\beta) := \text{enum}[\{\gamma \in \text{AP} \mid \forall \alpha' \in \alpha, \varphi(\alpha',\gamma) = \gamma\}](\beta) \end{eqnarray*} and $$\Phi \colon \text{On} \times \text{On} \to \text{On}$$ is defined as \begin{eqnarray*} \Phi(\alpha,\beta) := \text{enum}[\{\gamma \in \text{Card} \mid \forall \alpha' \in \alpha, \Phi(\alpha',\gamma) = \gamma\}](\beta) \end{eqnarray*} by the transfinite recursion on $$\alpha$$.

## 変種

A similar method is applicable to non-closed classes. For example, the class $$\textrm{Reg}$$ of regular cardinals is not closed, but the higher regularlity such as weak inaccessibility, weak 1-inaccessibility, and weak hyper-inaccessiblity is defined by using the enumerations of the closures $$\text{cl}(X)$$ of non-closed classes $$X$$. For example, the class of weakly inaccessible cardinals can be characterised as the class of fixed points of the enumeration function of $$\textrm{Reg}$$, or equivalently, the intersection of $$\textrm{Reg}$$ and the derivative of the enumeration $$\alpha \mapsto \aleph_{\alpha}$$ of the closure of $$\textrm{Reg}$$, i.e. the class of infinite cardinals. The class of weakly 1-inaccessible cardinals can be characterised as the class of fixed points of the enumeration function of the class of weakly inaccessible cardinals, or equivalently, the intersection of $$\textrm{Reg}$$ and the derivative of the enumeration $$\alpha \mapsto I_{\alpha}$$ of the closure of the class of weakly inaccessible cardinals. Similarly to $$\varphi$$ and $$\Phi$$, the $$2$$-ary enumeration function $$I(\alpha,\beta)$$ is defined as \begin{eqnarray*} I(\alpha,\beta) := \text{enum}[\text{cl}(\{\gamma \in \text{Reg} \mid \forall \alpha' \in \alpha, I(\alpha',\gamma) = \gamma\})](\beta) \end{eqnarray*} (or the variant given by replacing $$\text{Reg}$$ by the class of weakly inaccessible cardinals depending on the convention of the index) by the transfinite recursion on $$\alpha$$.

## 参考文献

1. A. Freund and M. Rathjen, Derivatives of normal functions in reverse mathematics. Annals of Pure and Applied Logic 172(2) 102890, 2021 (PDF on arxiv)
2. M. Rathjen, Ordinal notations based on a weakly Mahlo cardinal. Archive for Mathematical Logic 29(4) 249-263 (1990).