User blog comment:P進大好きbot/Proposal to Choose an Official Standard OCF in This Wiki/@comment-37485018-20191221014123

Is it true that \(\psi_0(\psi_1(\psi_2(\psi_3(0))))\neq\psi_0(\psi_3(0))\)? I thought that it went like this: \(\psi_0(\psi_3(0))\) can use \(\psi_2(\eta)\) for \(\eta<\psi_3(0)\), so \(\psi_0(\psi_3(0))=\sup(\{\psi_0(\psi_2(\eta)):\eta<\psi_3(0)\})=\psi_0(\psi_2(\psi_3(0)))\). Then \(\psi_2(\psi_3(0))=\varepsilon_{\Omega_2+1}\), so \(\psi_0(\psi_3(0))=\psi_0(\varepsilon_{\Omega_2+1})\). Then \(\psi_0(\varepsilon_{\Omega_2+1})\) can use \(\psi_1(\eta)\) for \(\eta<\varepsilon_{\Omega_2+1}\), so \(\psi_0(\varepsilon_{\Omega_2+1})=\sup(\{\psi_0(\psi_1(\eta)):\eta<\varepsilon_{\Omega_2+1}\})=\psi_0(\psi_1(\varepsilon_{\Omega_2+1}))\).