大數學 维基
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這是我的日文日記的中文翻譯。

縮寫

為了簡化解析表中的約定,我引入了\(T\)和\(DT\)的縮寫。

縮寫前 縮寫後
\(()\)
\(0\)
\(\textbf{四}(s,t)\)
(\((s,t) \in T^2\))
\(\textbf{四}_s(t)\)
\(\textbf{四}_0(t)\)
(\(t \in T\))
\(\textbf{四}(t)\)
\(\textbf{四}(0)\)
\(1\)
\(1+1\)
\(2\)
\(2+1\)
\(3\)
\(\textbf{四}(1)\)
\(\omega\)
\(\textbf{四}_1(0)\)
\(\Omega\)
\((2,v),s_0,s_1\)
(\((v,s_0,s_1) \in T^3)\)
\(v,s_0,s_1\)
\((3,v),s_0,s_1\)
(\((v,s_0,s_1) \in T^3)\)
\(v,0,s_0,s_1\)
\((u_0,v_0),\ldots,(u_n,v_n),s_0,s_1\)
(\(n \in \mathbb{N}, (u_0,\ldots,u_n,v_0,\ldots,v_n,s_0,s_1) \in T^{2n+4})\)
\(\begin{array}{ccccc} v_0 & \cdots & v_n & s_0 & s_1 \\ u_0 & \cdots & u_n & 1 & 0 \end{array}\)
\((1,0,1),s_0,s_1\)
(\((s_0,s_1) \in T^2)\)
\(\begin{array}{cccccc} 1 & s_0 & s_1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array}\)
\(\textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right)\)
\(M\)
\((1,0,1),(u_0,v_0).\ldots,(u_n,v_n),s_0,s_1\)
(\(n \in \mathbb{N}, (u_0,\ldots,u_n,v_0,\ldots,v_n,s_0,s_1) \in T^{2n+5})\)
\(\begin{array}{cccccc} 1 & v_0 & \cdots & v_n & s_0 & s_1 \\ 0 & u_0 & \cdots & u_n & 1 & 0 \\ 1 & 0 & \cdots & 0 & 0 & 0 \end{array}\)


解析

雖然沒有證據的線索,但我希望\((OT,<)\)形成一個序數符號。我列出了我的期待的\(t \in T\)的序型\(\textrm{ot}(t) \in \omega_1^{\textrm{CK }}\)。此後,\(\psi\)表示基於最弱的Mahlo基數的Rathjen的\(\psi\)。

\(t \in OT\) \(\textrm{ot}(t) \in \omega_1^{\textrm{CK}}\)
\(0\)
\(0\)
\(1\)
\(1 := \varphi_0(0)\)
\(2\)
\(2 := 1+1\)
\(3\)
\(3 := 2+1\)
\(\omega\)
\(\omega := \varphi_0(1)\)
\(\textbf{四}(\omega)\)
\(\varphi_0(\omega)\)
\(\textbf{四}(\Omega)\)
\(\varphi_1(0)\)
\(\textbf{四}(\Omega+\Omega)\)
\(\varphi_1(1)\)
\(\textbf{四}(\textbf{四}_1(1))\)
\(\varphi_1(\omega)\)
\(\textbf{四}(\textbf{四}_1(\Omega))\)
\(\varphi_2(0)\)
\(\textbf{四}(\textbf{四}_1(\textbf{四}_1(1)))\)
\(\varphi_{\omega}(0)\)
\(\textbf{四}(\textbf{四}_1(\textbf{四}_1(\Omega)))\)
\(\psi_{\chi_0(0)}(0)\)
\(\textbf{四}(\textbf{四}_2(0))\)
\(\psi_{\chi_0(0)}(\varphi_1(\chi_0(0)+1))\)
\(\textbf{四}(\textbf{四}_3(0))\)
\(\psi_{\chi_0(0)}(\varphi_1(\chi_0(1)+1))\)
\(\textbf{四}(\textbf{四}_{\omega}(0))\)
\(\psi_{\chi_0(0)}(\chi_0(\omega))\)
\(\textbf{四}(\textbf{四}_{\omega}(1))\)
\(= \textbf{四}(\textbf{四}(\textbf{季} 0))\)
\(\psi_{\chi_0(0)}(\varphi_0(\chi_0(\omega)+1))\)
\(\textbf{四}(\textbf{四}(1,0,0))\)
\(\psi_{\chi_0(0)}(\Phi_1(0))\)
\(\textbf{四}(\textbf{四}(2,0,0))\)
\(\psi_{\chi_0(0)}(\Phi_1(\chi_1(0)+1))\)
\(\textbf{四}(\textbf{四}(3,0,0))\)
\(\psi_{\chi_0(0)}(\Phi_1(\chi_1(1)+1))\)
\(\textbf{四}(\textbf{四}(1,0,0,0))\)
\(\psi_{\chi_0(0)}(\psi_{\chi_2(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ \omega & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\omega}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{ccc} 1 & \omega & 1 \\ \omega & 1 & 0 \end{array} \right) \right)\)
\(= \textbf{四}(\textbf{四}(\textbf{季} 1))\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\omega}(0)}(0) + \varphi_0(\chi_{\omega}(0)+1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ \Omega & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\chi_0(0)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ \textbf{四}_2(0) & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\psi_{\varphi_1(\chi_0(0)+1)}(0)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ \textbf{四}_{\omega}(0) & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\psi_{\chi_{\omega}(0)}(0)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ \textbf{四}(1,0,0) & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\Phi_1(0)}(0)}(0))\)
\(\textbf{四}(M)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M}(0)}(0))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{M}(0)}(0))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 1 \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_0(\psi_{\chi_{M}(0)}(0)+1))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_0(\psi_{\chi_{M}(0)}(0)+\psi_{\chi_{M}(0)}(0)))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & 0 \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_1(\psi_{\chi_{M}(0)}(0)+1))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & 0 \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_2(\psi_{\chi_{M}(0)}(0)+1))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \omega \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_{\omega}(\psi_{\chi_{M}(0)}(0)+1))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & 0 \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_0(\chi_{M}(0)+1)}(0))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & \textbf{四} \left( \begin{array}{ccc} 1 & 1 & 0 \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_0(\chi_{M}(0)+1)}(0))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 2 & 0 & 0 \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M}(0)}(1))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} \omega & 0 & 0 \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M}(0)}(\omega))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M & 1 & 0 \end{array} \right) & 0 & 0 \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M}(0)}(\psi_{\chi_{M}(0)}(0)))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M+1 & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\chi_{M}(0))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M+2 & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M+1}(0)}(0))\)
\(\textbf{四} \left( M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M + \textbf{四} \left( \begin{array}{ccc} 1 & 0 & 0 \\ M & 1 & 0 \end{array} \right) & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M+\psi_{\chi_{M}(0)}(0)}(0)}(0))\)
\(\textbf{四}(M+M)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{M+M}}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_0(M+1)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & M \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_0(M+M)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_0(\varphi_0(M+1))}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_1(M+1)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_2(M+1)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_{\omega}(M+1)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & M \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_M(1)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & M+1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\psi_{\chi_{\varphi_{M+1}(0)}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(M)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + 1 \right)\)
\(\varphi_0(\psi_{\chi_0(0)}(M)+1)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \Omega \right)\)
\(\varphi_1(\psi_{\chi_0(0)}(M)+1)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\Omega) \right)\)
\(\varphi_2(\psi_{\chi_0(0)}(M)+1)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,1)) \right)\)
\(\varphi_{\omega}(\psi_{\chi_0(0)}(M)+1)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\Omega)) \right)\)
\(\psi_{\chi_0(0)}(M+1)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\Omega)+1) \right)\)
\(\psi_{\chi_0(0)}(M+\omega)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\Omega)+\Omega) \right)\)
\(\psi_{\chi_0(0)}(M+\chi_0(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\Omega)+\textbf{四}(1,\Omega)) \right)\)
\(\psi_{\chi_0(0)}(M+\varphi_0(\chi_0(0)+\chi_0(0)))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\Omega+1)) \right)\)
\(\psi_{\chi_0(0)}(M+\varphi_0(\varphi_0(\chi_0(0)+1)))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\Omega+\Omega)) \right)\)
\(\psi_{\chi_0(0)}(M+\varphi_0(\varphi_0(\chi_0(0)+\chi_0(0))))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(1,\textbf{四}(1,1))) \right)\)
\(\psi_{\chi_0(0)}(M+\varphi_0(\varphi_0(\varphi_0(\chi_0(0)+1))))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,\textbf{四}(2,0)) \right)\)
\(\psi_{\chi_0(0)}(M+\varphi_1(\chi_0(0)+1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(2,0) \right)\)
\(\psi_{\chi_0(0)}(M+\chi_0(1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四}(1,0,0) \right)\)
\(\psi_{\chi_0(0)}(M+\chi_1(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 0 \\ \omega & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(M+\psi_{\chi_{\omega}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 0 \\ M & 1 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(M+\psi_{\chi_{M}(0)}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + M \right)\)
\(\psi_{\chi_0(0)}(M+M)\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_0(M+1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & M \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_0(M+M))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_0(\varphi_0(M+1)))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_1(M+1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_2(M+1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_{\omega}(M+1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & M \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_{M}(1))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & M+1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_{M+1}(0))\)
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) + \textbf{四} \left( \begin{array}{cccccc} 1 & 0 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
\(\psi_{\chi_0(0)}(\varphi_{\varphi_{\cdot_{\cdot_{\cdot_{\varphi_{M+1}(0)}\cdot}\cdot}\cdot}(0)}(0))\)
\(=\) \(\psi\)的可算限界
\(\textbf{四} \left( \textbf{四} \left( \begin{array}{cccccc} 1 & 1 & 0 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{array} \right) \right)\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,1),2,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,1),\omega,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,1),(2,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,1),(\omega,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,1),(M,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,1),(\textbf{四}(((2,1),0,1),(M,1),0,0),1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),0,2),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,1),1,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((2,2),0,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((3,1),0,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((\omega,1),0,1),0,0))\)
 ?
\(\textbf{四}(\textbf{四}(((\omega,1),\omega,1),\omega,1))\)
\(= \textbf{四}(\textbf{四}(\textbf{季} 2))\)
 ?
\(\textbf{四}(\textbf{四}((((\omega,1),\omega,1),\omega,1),\omega,1))\)
\(= \textbf{四}(\textbf{四}(\textbf{季} 3))\)
 ?
\(\Omega\)
 ?
\(\textbf{四}(\omega,1)\)
\(= \textbf{四}(\textbf{季} 0)\)
 ?
\(\textbf{四} \left( \begin{array}{ccc} 1 & \omega & 1 \\ \omega & 1 & 0 \end{array} \right)\)
\(= \textbf{四}(\textbf{季} 1)\)
 ?
\(\textbf{四}(((\omega,1),\omega,1),\omega,1)\)
\(= \textbf{四}(\textbf{季} 2)\)
 ?
\(\textbf{四}((((\omega,1),\omega,1),\omega,1),\omega,1)\)
\(= \textbf{四}(\textbf{季} 3)\)
 ?
\(\textbf{四} \underbrace{(((((\cdots),\omega,1),\omega,1),\omega,1),\omega,1)}_{n+1}\) (\(n \in \mathbb{N}\))
\(= \textbf{四}(\textbf{季} n)\)
 ?
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