時々よく分からなくなるのでここにまとめておきます。
間違いがあればコメントで教えてください。
0~\(\varphi(2,0)\)[]
\(0=0\)
\(\psi_0(0)=1\)
\(\psi_0(0)+\psi_0(0)=2\)
\(\psi_0(\psi_0(0))=\omega\)
\(\psi_0(\psi_0(0))+\psi_0(\psi_0(0))=\omega \times 2\)
\(\psi_0(\psi_0(0)+\psi_0(0))=\omega^2\)
\(\psi_0(\psi_0(\psi_0(0)))=\omega^\omega\)
\(\psi_0(\psi_1(0))=\varepsilon_0\)
\(\psi_0(\psi_1(0)+\psi_0(0))=\varepsilon_0 \times \omega\)
\(\psi_0(\psi_1(0)+\psi_0(0)+\psi_(0))=\varepsilon_0 \times \omega^2\)
\(\psi_0(\psi_1(0)+\psi_0(\psi_0(0)))=\varepsilon_0 \times \omega^\omega\)
\(\psi_0(\psi_1(0)+\psi_0(\psi_1(0)))={\varepsilon_0}^2\)
\(\psi_0(\psi_1(0)+\psi_0(\psi_1(0)+\psi_0(\psi_1(0))))={\varepsilon_0}^{\varepsilon_0}\)
\(\psi_0(\psi_1(0)+\psi_1(0))=\varepsilon_1\)
\(\psi_0(\psi_1(0)+\psi_1(0)+\psi_1(0))=\varepsilon_2\)
\(\psi_0(\psi_1(\psi_0(0)))=\varepsilon_\omega\)
\(\psi_0(\psi_1(\psi_0(0)+\psi_0(0)))=\varepsilon_{\omega^2}\)
\(\psi_0(\psi_1(\psi_0(0)+\psi_0(0)+\psi_0(0)))=\varepsilon_{\omega^\omega}\)
\(\psi_0(\psi_1(\psi_0(\psi_1(0))))=\varepsilon_{\varepsilon_0}\)
\(\psi_0(\psi_1(\psi_1(0)))=\varphi(2,0)\)