集合論において微分 (derivative) とは順序数のクラス \(\textrm{On}\) を対角化する基本的な概念である。解析学の微分とは異なるので混同しないこと。

定義

正規関数クリーネの不動点定理を拡張した重要な性質を持つ。すなわち、正則関数はいくらでも大きな不動点を持つというものである。順序数の関数 \(f\) の微分 \(f'\) は \(f\) の不動点を数える関数として定義され[1]、すべての正則関数は微分可能である。それだけではなく、正則関数の微分は正則関数である。したがって、すべての正則関数は2回微分可能、3回微分可能であり、そのような操作は超限的に繰り返すことが可能なため(低階微分の共通する不動点をとる)、常に正則関数が得られる。

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 \(\alpha\).[2]

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).
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