FANDOM


ここでは急増加関数によく使用する超限順序数について、簡単な解説を含めて並べます。
どれがどれより大きいのか、というのをすぐ忘れてしまう人の為に、そういうものがあったらいいな、ということで作ってあります。
φ関数の定義はこちら

順序数 解説 基本列
\(\omega\) 最小の超限順序数かつ最小の極限順序数かつ最小の許容順序数 \(0, 1, 2, 3, \cdots\)
\(\omega \times 2 = \omega + \omega\) \(\omega, \omega+1, \omega+2, \omega+3, \cdots\)
\(\omega \times 3 = \omega \times 2 + \omega\) \(\omega \times 2, \omega \times 2 +1, \omega \times 2 +2, \omega \times 2 +3, \cdots\)
\(\omega^2 = \omega \times \omega\) \(0, \omega, \omega \times 2, \omega \times 3, \omega \times 4, \cdots\)
\(\omega^2+\omega\) \(\omega^2,\omega^2+1,\omega^2+2,\omega^2+3,\cdots\)
\(\omega^2\times2\) \(\omega^2,\omega^2+\omega,\omega^2+\omega\times2,\omega^2+\omega\times3,\cdots\)
\(\omega^3 = \omega^2 \times \omega\) \(0, \omega^2, \omega^2 \times 2, \omega^2 \times 3, \omega^2 \times 4, \cdots\)
\({^{2}\omega} = \omega^\omega = \omega \uparrow \uparrow 2\) \(1, \omega, \omega^2, \omega^3, \omega^4, \cdots\)
\(\omega^{\omega+1} = \omega^{\omega} \cdot \omega\) \(0, \omega^{\omega}, \omega^{\omega} \cdot 2, \omega^{\omega} \cdot 3, \omega^{\omega} \cdot 4, \cdots\)
\(\omega^{\omega+2}\) \(0,\omega^{\omega+1},\omega^{\omega+1}\times2,\omega^{\omega+1}\times3,\cdots\)
\(\omega^{\omega\times2}\) \(\omega^\omega,\omega^{\omega+1},\omega^{\omega+2},\omega^{\omega+3},\cdots\)
\(\omega^{\omega^2}\) \(1,\omega^\omega,\omega^{\omega\times2},\omega^{\omega\times3},\cdots\)
\({^{3}\omega} = \omega^{\omega^\omega} = \omega \uparrow \uparrow 3\) \(\omega, \omega^\omega, \omega^{\omega^2}, \omega^{\omega^3}, \omega^{\omega^4}, \cdots\)
\(\varepsilon_0\)\( = \omega \uparrow \uparrow \omega\) \(\alpha = \omega^\alpha\) が成り立つ最小の順序数 \(1, \omega, \omega^\omega, \omega^{\omega^\omega}, \omega^{\omega^{\omega^\omega}}, \cdots\)
\(\varepsilon_0\times2\) \(\varepsilon_0+1,\varepsilon_0+\omega,\varepsilon_0+\omega^\omega,\varepsilon_0+\omega^{\omega^\omega},\cdots\)
\(\varepsilon_0\times\omega\) \(0,\varepsilon_0,\varepsilon_0\times2,\varepsilon_0\times3,\cdots\)
\(\varepsilon_0^2\) \(\varepsilon_0,\varepsilon_0\times\omega,\varepsilon_0\times\omega^\omega,\varepsilon_0\times\omega^{\omega^\omega},\cdots\)
\(\varepsilon_0^\omega\) \(1,\varepsilon_0,\varepsilon_0^2,\varepsilon_0^3,\cdots\)
\(\varepsilon_0^{\varepsilon_0}\) \(\varepsilon_0,\varepsilon_0^\omega,\varepsilon_0^{\omega^\omega},\varepsilon_0^{\omega^{\omega^\omega}},\cdots\)
\(\varepsilon_1 = {^{\omega}\varepsilon_0}\) \(\alpha = \omega^\alpha\) が成り立つ 2 番目の順序数(最初を0番目と数えて、1番目と書かれることも多い) \(1, \varepsilon_0, \varepsilon_0^{\varepsilon_0}, \varepsilon_0^{\varepsilon_0^{\varepsilon_0}}, \varepsilon_0^{\varepsilon_0^{\varepsilon_0^{\varepsilon_0}}}, \cdots\)

または \(\varepsilon_0+1, \omega^{\varepsilon_0+1}, \omega^{\omega^{\varepsilon_0+1}}, \omega^{\omega^{\omega^{\varepsilon_0+1}}}, \cdots\)

\(\varepsilon_2 = {^{\omega}\varepsilon_1}\) \(\alpha = \omega^\alpha\) が成り立つ 3 番目の順序数 \(1, \varepsilon_1, \varepsilon_1^{\varepsilon_1}, \varepsilon_1^{\varepsilon_1^{\varepsilon_1}}, \varepsilon_1^{\varepsilon_1^{\varepsilon_1^{\varepsilon_1}}}, \cdots\)
\(\varepsilon_\omega\) \(\alpha = \omega^\alpha\) が成り立つ \(\omega\) 番目の順序数 \(\varepsilon_0, \varepsilon_1, \varepsilon_2, \varepsilon_3, \cdots\)
\(\varepsilon_{\omega+1} = {^{\omega}\varepsilon_\omega}\) \(\alpha = \omega^\alpha\) が成り立つ \(\omega+1\) 番目の順序数 \(1, \varepsilon_\omega, \varepsilon_\omega^{\varepsilon_\omega}, \varepsilon_\omega^{\varepsilon_\omega^{\varepsilon_\omega}}, \varepsilon_\omega^{\varepsilon_\omega^{\varepsilon_\omega^{\varepsilon_\omega}}}, \cdots\)

または \(\varepsilon_\omega+1, \omega^{\varepsilon_\omega+1}, \omega^{\omega^{\varepsilon_\omega+1}}, \omega^{\omega^{\omega^{\varepsilon_\omega+1}}}, \cdots\)

\(\varepsilon_{\omega\times2}\) \(\alpha = \omega^\alpha\) が成り立つ \(\omega\times2\) 番目の順序数 \(\varepsilon_{\omega},\varepsilon_{\omega+1},\varepsilon_{\omega+2},\varepsilon_{\omega+3},\cdots\)
\(\varepsilon_{\omega^2}\) \(\varepsilon_0,\varepsilon_\omega,\varepsilon_{\omega\times2},\varepsilon_{\omega\times3},\cdots\)
\(\varepsilon_{\omega^\omega}\) \(\varepsilon_1,\varepsilon_\omega,\varepsilon_{\omega^2},\varepsilon_{\omega^3},\cdots\)
\(\varepsilon_{\varepsilon_0}\) \(\alpha = \omega^\alpha\) が成り立つ \(\varepsilon_0\) 番目の順序数 \(\varepsilon_1, \varepsilon_\omega, \varepsilon_{\omega^{\omega}}, \varepsilon_{\omega^{\omega^{\omega}}}, \varepsilon_{\omega^{\omega^{\omega^{\omega}}}}, \cdots\)
\(\phi\)\((2,0) = \eta_0\) (GWiki では \(\zeta_0\)) \(\alpha = \varepsilon_\alpha\) が成り立つ最小の順序数 \(0, \varepsilon_0, \varepsilon_{\varepsilon_0}, \varepsilon_{\varepsilon_{\varepsilon_0}}, \varepsilon_{\varepsilon_{\varepsilon_{\varepsilon_0}}}, \cdots\)
\(\phi(2,0)^2\) \(0,\phi(2,0)\times\varepsilon_0,\phi(2,0)\times\varepsilon_{\varepsilon_0},\phi(2,0)\times\varepsilon_{\varepsilon_{\varepsilon_0}},\cdots\)
\(\phi(2,0)^\omega\) \(1,\phi(2,0),\phi(2,0)^2,\phi(2,0)^3,\cdots\)
\(\phi(2,0)^{\varepsilon_0}\) \(\phi(2,0),\phi(2,0)^\omega,\phi(2,0)^{\omega^\omega},\phi(2,0)^{\omega^{\omega^\omega}},\cdots\)
\(\phi(2,0)^{\phi(2,0)}\) \(1,\phi(2,0)^{\varepsilon_0},\phi(2,0)^{\varepsilon_{\varepsilon_0}},\phi(2,0)^{\varepsilon_{\varepsilon_{\varepsilon_0}}},\cdots\)
\(\varepsilon_{\phi(2,0)+1}\) \(1,\phi(2,0),\phi(2,0)^{\phi(2,0)},\phi(2,0)^{\phi(2,0)^{\phi(2,0)}},\cdots\)
\(\varepsilon_{\phi(2,0)+2}\) \(1,\varepsilon_{\phi(2,0)+1},\varepsilon_{\phi(2,0)+1}^{\varepsilon_{\phi(2,0)+1}},\varepsilon_{\phi(2,0)+1}^{\varepsilon_{\phi(2,0)+1}^{\varepsilon_{\phi(2,0)+1}}},\cdots\)
\(\varepsilon_{\phi(2,0)+\omega}\) \(\phi(2,0),\varepsilon_{\phi(2,0)+1},\varepsilon_{\phi(2,0)+2},\varepsilon_{\phi(2,0)+3},\cdots\)
\(\varepsilon_{\phi(2,0)+\omega\times2}\) \(\varepsilon_{\phi(2,0)+\omega},\varepsilon_{\phi(2,0)+\omega+1},\varepsilon_{\phi(2,0)+\omega+2},\varepsilon_{\phi(2,0)+\omega+3},\cdots\)
\(\varepsilon_{\phi(2,0)\times2}\) \(\varepsilon_{\phi(2,0)},\varepsilon_{\phi(2,0)+\varepsilon_0},\varepsilon_{\phi(2,0)+\varepsilon_{\varepsilon_0}},\varepsilon_{\phi(2,0)+\varepsilon_{\varepsilon_{\varepsilon_0}}},\cdots\)
\(\varepsilon_{\phi(2,0)\times\omega}\) \(\varepsilon_0,\phi(2,0),\varepsilon_{\phi(2,0)\times2},\varepsilon_{\phi(2,0)\times3},\cdots\)
\(\varepsilon_{\phi(2,0)^2}\) \(\varepsilon_1,\varepsilon_{\phi(2,0)\times\varepsilon_0},\varepsilon_{\phi(2,0)\times\varepsilon_{\varepsilon_0}},\varepsilon_{\phi(2,0)\times\varepsilon_{\varepsilon_{\varepsilon_0}}},\cdots\)
\(\varepsilon_{\phi(2,0)^{\phi(2,0)}}\) \(\varepsilon_1,\varepsilon_{\phi(2,0)^{\varepsilon_0}},\varepsilon_{\phi(2,0)^{\varepsilon_{\varepsilon_0}}}\varepsilon_{\phi(2,0)^{\varepsilon_{\varepsilon_{\varepsilon_0}}}},\cdots\)
\(\varepsilon_{\varepsilon_{\phi(2,0)+1}}\) \(\varepsilon_1,\phi(2,0),\varepsilon_{\phi(2,0)^{\phi(2,0)}},\varepsilon_{\phi(2,0)^{\phi(2,0)^{\phi(2,0)}}},\cdots\)
\(\phi(2,1) = \eta_1\) (\(\zeta_1\)) \(\alpha = \varepsilon_\alpha\) が成り立つ2番目の順序数 \(\phi(2,0)+1, \varepsilon_{\phi(2,0)+1}, \varepsilon_{\varepsilon_{\phi(2,0)+1}}, \cdots\)
\(\varepsilon_{\phi(2,1)+1}\) \(1,\phi(2,1),\phi(2,1)^{\phi(2,1)},\phi(2,1)^{\phi(2,1)^{\phi(2,1)}},\cdots\)
\(\phi(2,2) = \eta_2\) (\(\zeta_2\)) \(\alpha = \varepsilon_\alpha\) が成り立つ3番目の順序数 \(\phi(2,1)+1, \varepsilon_{\phi(2,1)+1}, \varepsilon_{\varepsilon_{\phi(2,1)+1}}, \cdots\)
\(\phi(2,\omega) = \eta_\omega\) (\(\zeta_\omega\)) \(\alpha = \varepsilon_\alpha\) が成り立つ\(\omega\)番目の順序数 \(\phi(2,0), \phi(2,1), \phi(2,2), \phi(2,3), \cdots\)
\(\varepsilon_{\phi(2,\omega)+1}\) \(1,\phi(2,\omega),\phi(2,\omega)^{\phi(2,\omega)},\phi(2,\omega)^{\phi(2,\omega)^{\phi(2,\omega)}},\cdots\)
\(\phi(2,\omega+1)\) \(\phi(2,\omega)+1,\varepsilon_{\phi(2,\omega)+1},\varepsilon_{\varepsilon_{\phi(2,\omega)+1}},\varepsilon_{\varepsilon_{\varepsilon_{\phi(2,\omega)+1}}},\cdots\)
\(\phi(2,\phi(2,0))\) \(\phi(2,0),\phi(2,\varepsilon_0),\phi(2,\varepsilon_{\varepsilon_0}),\phi(2,\varepsilon_{\varepsilon_{\varepsilon_0}}),\cdots\)
\(\phi(3,0)\) \(\alpha = \phi(2,\alpha)\) が成り立つ最小の順序数 \(0,\phi(2,0), \phi(2,\phi(2,0)), \phi(2,\phi(2,\phi(2,0)), \cdots\)
\(\varepsilon_{\phi(3,0)+1}\) \(1,\phi(3,0),\phi(3,0)^{\phi(3,0)},\phi(3,0)^{\phi(3,0)^{\phi(3,0)}},\cdots\)
\(\phi(2,\phi(3,0)+1)\) \(\phi(3,0)+1,\varepsilon_{\phi(3,0)+1}.\varepsilon_{\varepsilon_{\phi(3,0)+1}},\varepsilon_{\varepsilon_{\varepsilon_{\phi(3,0)+1}}},\cdots\)
\(\phi(3,1)\) \(\alpha = \phi(2,\alpha)\) が成り立つ2番目の順序数 \(\phi(3,0)+1,\phi(2,\phi(3,0)+1),\phi(2,\phi(2,\phi(3,0)+1)),\phi(2,\phi(2,\phi(2,\phi(3,0)+1))),\cdots\)
\(\phi(3,\omega)\) \(\alpha = \phi(2,\alpha)\) が成り立つ\(\omega\)番目の順序数 \(\phi(3,0),\phi(3,1),\phi(3,2),\phi(3,3),\cdots\)
\(\phi(3,\phi(3,0))\) \(\alpha = \phi(2,\alpha)\) が成り立つ\(\phi(3,0)\)番目の順序数 \(\phi(3,0),\phi(3,\phi(2,0)),\phi(3,\phi(2,\phi(2,0))),\phi(3,\phi(2,\phi(2,\phi(2,0)))),\cdots\)
\(\phi(4,0)\) \(\alpha = \phi(3,\alpha)\) が成り立つ最小の順序数 \(0,\phi(3,0),\phi(3,\phi(3,0)),\phi(3,\phi(3,\phi(3,0))),\cdots\)
\(\phi(\omega,0)\) 任意の非負整数mに対して\(\alpha = \phi(m,\alpha)\) が成り立つ最小の順序数 \(\phi(0,0), \phi(1,0), \phi(2,0), \phi(3,0), \cdots\)
\(\phi(\omega,1)\) 任意の非負整数mに対して\(\alpha = \phi(m,\alpha)\) が成り立つ2番目の順序数 \(\phi(\omega,0)+1, \phi(0,\phi(\omega,0)+1), \phi(1,\phi(\omega,0)+1), \phi(2,\phi(\omega,0)+1), \cdots\)
\(\phi(1,0,0)=\vartheta(\Omega)=\Gamma_0\) フェファーマン・シュッテの順序数。

\(\alpha=\phi(\alpha,0)\)が成り立つ最小の順序数。\(\Omega\)は通常\(\omega_1\)とする。

\(1, \varepsilon_0, \phi(\varepsilon_0,0), \phi(\phi(\varepsilon_0,0),0), \cdots\)

または \(\phi(0,0),\phi(\phi(0,0),0),\phi(\phi(\phi(0,0),0),0),\cdots\)

\(\phi(1,0,1)=\Gamma_1\) \(\alpha=\phi(\alpha,0)\)が成り立つ2番目の順序数 \(\Gamma_0+1, \phi(\Gamma_0+1,0), \phi(\phi(\Gamma_0+1,0),0), \cdots\)
\(\phi(1,1,0)\) \(\alpha=\Gamma_\alpha\)が成り立つ最小の順序数 \(\Gamma_0, \Gamma_{\Gamma_0}, \Gamma_{\Gamma_{\Gamma_0}}, \cdots\)

または \(\phi(1,0,0),\phi(1,0,\phi(1,0,0)), \phi(1,0,\phi(1,0,\phi(1,0,0))), \cdots\)

\(\phi(1,2,0)\) \(\alpha=\phi(1,1,\alpha)\)が成り立つ最初の順序数 \(\phi(1,1,0) , \phi(1,1,\phi(1,1,0)), \phi(1,1,\phi(1,1,\phi(1,1,0))) , \cdots\)
\(\phi(2,0,0)\) \(\alpha=\phi(1,\alpha,0)\)が成り立つ最初の順序数

\(\Gamma_0, \phi(1,\Gamma_0,0), \phi(1,\phi(1,\Gamma_0,0),0),\cdots\)

\(\phi(1,0,0,0)=\vartheta(\Omega^2)\) アッカーマン順序数。

\(\alpha=\phi(\alpha,0,0)\)が成り立つ最小の順序数

\(1, \Gamma_0, \phi(\Gamma_0,0,0), \phi(\phi(\Gamma_0,0,0),0,0), \cdots\)
\(\phi(1,0,0,1)\) \(\alpha=\phi(\alpha,0,0)\)が成り立つ2番目の順序数  
\(\phi(1,0,1,0)\) \(\alpha=\phi(1,0,0,\alpha)\)が成り立つ最小の順序数 \(\phi(1,0,0,0), \phi(1,0,0,\phi(1,0,0,0)), \cdots\)
\(\phi(1,1,0,0)\) \(\alpha=\phi(1,0,\alpha,0)\)が成り立つ最小の順序数 \(\phi(1,0,0,0), \phi(1,0,\phi(1,0,0,0),0), \cdots\)
\(\phi(2,0,0,0)\) \(\alpha=\phi(1,\alpha,0,0)\)が成り立つ最初の順序数 \(\phi(1,0,0,0), \phi(1,\phi(1,0,0,0),0,0), \cdots\)
\(\phi(1,0,0,0,0)=\vartheta(\Omega^3)\) \(1, \phi(1,0,0,0), \phi(\phi(1,0,0,0),0,0,0), \cdots\)
\(\vartheta(\Omega^\omega)\) 小ヴェブレン順序数 (多変数ヴェブレン関数で表せない最小の順序数) \(\Gamma_0, \vartheta(\Omega^2), \vartheta(\Omega^3), \cdots\)
\(\vartheta(\Omega^{\vartheta(\Omega^\omega)})\)   \(\vartheta(\Omega^\omega), \vartheta(\Omega^{\varepsilon_0}), \vartheta(\Omega^{\varGamma_0}), \cdots\)
\(\vartheta(\Omega^\Omega)\) 大ヴェブレン順序数 (\(\alpha=\vartheta(\Omega^{\vartheta(\alpha)})\)が成り立つ最小の順序数) \(\varepsilon_0, \vartheta(\Omega^{\varepsilon_0}), \vartheta(\Omega^{\vartheta(\Omega^{\varepsilon_0})}), \cdots\)
\(\vartheta(\varepsilon_{\Omega+1})=\psi(\varepsilon_{\Omega+1})\) バッハマン・ハワード順序数 (\(\alpha=\vartheta(\Omega^\alpha)\)が成り立つ最小の順序数) \(\vartheta(0), \vartheta(1), \vartheta(\Omega), \vartheta(\Omega^\Omega), \vartheta(\Omega^{\Omega^\Omega}), \cdots\)
\(\vartheta(\varepsilon_{\Omega+\omega})\) \(\vartheta(\Omega), \vartheta(\varepsilon_{\Omega+1}), \vartheta(\varepsilon_{\Omega+2}), \vartheta(\varepsilon_{\Omega+3}), \cdots\)
\(\vartheta(\eta_{\Omega+1})\) (GWikiでは\(\vartheta(\zeta_{\Omega+1})\)) \(\alpha = \vartheta(\varepsilon_{\Omega+\alpha})\)が成り立つ最小の順序数 \(\vartheta(\varepsilon_{\Omega+1}), \vartheta(\varepsilon_{\Omega+\vartheta(\varepsilon_{\Omega+1})}), \vartheta(\varepsilon_{\Omega+\vartheta(\varepsilon_{\Omega+\vartheta(\varepsilon_{\Omega+1})})}), \cdots\)
\(\vartheta(\phi(\Omega,1))\) \(\alpha = \vartheta(\phi(\alpha, \Omega+1))\)が成り立つ最小の順序数 \(\vartheta(\varepsilon_{\Omega+1}), \vartheta(\phi(\vartheta(\varepsilon_{\Omega+1}), \Omega+1)), \cdots\)
\(\vartheta(\Omega_2) = \vartheta(\Gamma_{\Omega+1})\) \(\vartheta(\phi(\Omega,1)), \vartheta(\phi(\phi(\Omega,1),0)), \vartheta(\phi(\phi(\phi(\Omega,1),0),0)), \cdots\)
\({\vartheta({\Omega_{\omega}})}={\psi_0({\Omega_{\omega}})}\) \({\vartheta({\Omega})}, {\vartheta({\Omega_2})}, {\vartheta({\Omega_3})}, \cdots\)
\({\psi_0({\varepsilon_{\Omega_{\omega}+1}})}\) 竹内・フェファーマン・ブーフホルツ順序数 \({\psi_0({\Omega_{\omega}})},{\psi_0({\Omega_{\omega}^{\Omega_{\omega}}})},{\psi_0({\Omega_{\omega}^{\Omega_{\omega}^{\Omega_{\omega}}}})}, \cdots\)
\({\psi_0({\Omega_{\Omega}})}\) \({\alpha}={\psi_0({\Omega_{\alpha}})}\)が成り立つ最小の順序数。 \({\psi_0({\Omega})},{\psi_0({\Omega_{\psi_0({\Omega})}})},{\psi_0({\Omega_{\psi_0({\Omega_{\psi_0({\Omega})}})}})},\cdots\)
\({\psi_0({\psi_I(0)})}\) \({\psi_0({\Omega})},{\psi_0({\Omega_{\Omega}})},{\psi_0({\Omega_{\Omega_{\Omega}}})},\cdots\)
\(\omega^\text{CK}_1\) チャーチ・クリーネ順序数(2 番目の許容順序数)、自明な収束列は存在しない。
\(\omega_1=\aleph_1\) 最初の非可算な順序数。収束列は存在しない
\(\psi_I(0) \) Ω収束点(\(\alpha \mapsto \Omega_\alpha\)が成り立つ最小の順序数) \(\Omega, \Omega_\Omega, \Omega_{\Omega_\Omega} ,\cdots\)
\(\psi_{\Omega_{\psi_I(0)+1}}(0)\) \(\psi_I(0), \psi_I(0)^{\psi_I(0)}, \psi_I(0)^{\psi_I(0)^{\psi_I(0)}},\cdots\)
\(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+1})\) \(\alpha \mapsto \psi_{\Omega_{\psi_I(0)+1}}(\alpha)\) が成り立つ最小の順序数 \(0, \psi_{\Omega_{\psi_I(0)+1}}(0), \psi_{\Omega_{\psi_I(0)+1}}(\psi_{\Omega_{\psi_I(0)+1}}(0)),\cdots\)
\(\psi_{\Omega_{\psi_I(0)+1}}(\Omega_{\psi_I(0)+\omega})\) \(0, \psi_{\Omega_{\psi_I(0)+1}}(0), \psi_{\Omega_{\psi_I(0)+1}}(\psi_{\Omega_{\psi_I(0)+2}}(0)) ,\cdots\)
\(\psi_I(I)\) \(\alpha \mapsto \psi_I(\alpha)\) が成り立つ最小の順序数 \(0, \psi_I(0), \psi_I(\psi_I(0)), \cdots\)
\(\psi_I(\psi_{I_2}(0))\) \(\psi_I(\Omega_{I+1}), \psi_I(\Omega_{\Omega_{I+1}}), \psi_I(\Omega_{\Omega_{\Omega_{I+1}}})\)
\(\psi_I(\psi_{I_2}(I))\) \(\psi_I(\psi_{I_2}(0)), \psi_I(\psi_{I_2}(\psi_I(\psi_{I_2}(0)))), \psi_I(\psi_{I_2}(\psi_I(\psi_{I_2}(\psi_I(\psi_{I_2}(0)))))), \cdots\)
\(\psi_I(\psi_{I_2}(I_2))\) \(\psi_I(\psi_{I_2}(0)), (\psi_I(\psi_{I_2}(\psi_{I_2}(0))), \psi_I(\psi_{I_2}(\psi_{I_2}(0))), \cdots\)
\(\psi_{I_{\omega}}(0)\) \(I, I_2, I_3, \cdots\)
\(\psi_{\chi(1,0)}(0)\) \(I, I_I, I_{I_I}, \cdots\)
\(\chi(1,0)\) 最小の弱 1 - 到達不能基数
\(\psi_{\chi(M,0)}(0)\) \(\chi(0,0),\chi(\chi(0,0),0),\chi(\chi(\chi(0,0),0),0),\cdots\)
\(\chi(M,0)\) 最小の弱 hyper - 到達不可能基数
\(M\) 最小の弱マーロ基数
\(\Xi(K,0)\) 最小の弱 hyper - マーロ基数
\(\beth_1\) ZFC から、\(\beth_1=\aleph_1\) は連続体仮説と同値