| What you input in 'Ordinal' | What it represents |
|---|---|
| w^w | $\omega^\omega$ |
| e+1 | $\varepsilon_{0} + 1$ |
| v(2,w) | $ \varphi(2,\omega) $ |
| v(1,0,1) | $ \varphi(1,0,1) $ |
| v(1@w) | $ \varphi(1@\omega) $ |
| (work in progress) | $ \varphi(1@(1,0)) $ |
Note: invalid and ill-formed input (like $1 +\omega $) is not strictly checked. Correctness not guaranteed in this situation.
$$ f_{0}(3)=4 $$
$$ f_{1}(3)=6 $$
$$ f_{2}(3)=24 $$
$$ f_{3}(2)=2048 $$
$$ f_{3}(3)=f_{2}(402653184)=\dots $$
$$ f_{\omega}(3)=f_{2}(402653184)=\dots $$
$$ f_{\omega+1}(3)=f_{\omega}^{2}(f_{2}(402653184))=\dots $$
$$ \begin{align*} \omega\cdot2[3] &= \omega+\omega[3] \\ &= \omega+3 \\ \end{align*} $$
$$ f_{\omega\cdot2+1}(3)=f_{\omega\cdot2}^{2}(f_{\omega+2}^{2}(f_{\omega+1}^{2}(f_{\omega}^{2}(f_{2}(402653184)))))=\dots $$
$$ \begin{align*} \varphi(0, 2)[3] &= \omega\cdot\omega[3] \\ &= \omega\cdot3[3] \\ &= \omega\cdot2+\omega[3] \\ &= \omega\cdot2+3 \\ \end{align*} $$
$$ \omega^{2}[4]=\omega\cdot3+4 $$
$$ \omega^{2}+\omega[3]=\omega^{2}+3 $$
$$ f_{\omega^{2}+\omega+1}(3)=f_{\omega^{2}+\omega}^{2}(f_{\omega^{2}+2}^{2}(f_{\omega^{2}+1}^{2}(f_{\omega^{2}}^{2}(f_{\omega\cdot2+2}^{2}(f_{\omega\cdot2+1}^{2}(f_{\omega\cdot2}^{2}(f_{\omega+2}^{2}(f_{\omega+1}^{2}(f_{\omega}^{2}(f_{2}(402653184)))))))))))=\dots $$
$$ \omega^{3}[4]=\omega^{2}\cdot3+\omega\cdot3+4 $$
$$ \omega^{\omega}[3]=\omega^{2}\cdot2+\omega\cdot2+3 $$
$$ f_{\omega^{\omega}}(3)=f_{\omega^{2}\cdot2+\omega\cdot2+3}(3)=\dots $$
$$ \begin{align*} \varphi(0, \omega+1)[3] &= \omega^{\omega}\cdot\omega[3] \\ &= \omega^{\omega}\cdot3[3] \\ &= \omega^{\omega}\cdot2+\omega^{\omega}[3] \\ &= \omega^{\omega}\cdot2+\omega^{\omega[3]} \\ &= \omega^{\omega}\cdot2+\omega^{3}[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot\omega[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot3[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega^{2}[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega\cdot\omega[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega\cdot3[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega\cdot2+\omega[3] \\ &= \omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega\cdot2+3 \\ \end{align*} $$
$$ f_{\omega^{\omega}+1}(3)=f_{\omega^{\omega}}^{2}(f_{\omega^{2}\cdot2+\omega\cdot2+2}^{2}(f_{\omega^{2}\cdot2+\omega\cdot2+1}^{2}(f_{\omega^{2}\cdot2+\omega\cdot2}^{2}(f_{\omega^{2}\cdot2+\omega+2}^{2}(f_{\omega^{2}\cdot2+\omega+1}^{2}(f_{\omega^{2}\cdot2+\omega}^{2}(f_{\omega^{2}\cdot2+2}^{2}(f_{\omega^{2}\cdot2+1}^{2}(f_{\omega^{2}\cdot2}^{2}(f_{\omega^{2}+\omega\cdot2+2}^{2}(f_{\omega^{2}+\omega\cdot2+1}^{2}(f_{\omega^{2}+\omega\cdot2}^{2}(f_{\omega^{2}+\omega+2}^{2}(f_{\omega^{2}+\omega+1}^{2}(f_{\omega^{2}+\omega}^{2}(f_{\omega^{2}+2}^{2}(f_{\omega^{2}+1}^{2}(f_{\omega^{2}}^{2}(f_{\omega\cdot2+2}^{2}(f_{\omega\cdot2+1}^{2}(f_{\omega\cdot2}^{2}(f_{\omega+2}^{2}(f_{\omega+1}^{2}(f_{\omega}^{2}(f_{2}(402653184))))))))))))))))))))))))))=\dots $$
$$ \varphi(0, \omega\cdot2)[3]=\omega^{\omega+2}\cdot2+\omega^{\omega+1}\cdot2+\omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega\cdot2+3 $$
$$ \varphi(1, 0)[0]=0 $$
$$ \varphi(1, 0)[1]=1 $$
$$ \begin{align*} \varphi(1, 0)[2] &= \omega^{\varphi(1, 0)[1]} \\ &= \omega^{\omega^{\varphi(1, 0)[0]}} \\ &= \omega^{\omega^{0}[2]} \\ &= \omega^{1}[2] \\ &= \omega[2] \\ &= 2 \\ \end{align*} $$
$$ \begin{align*} \varepsilon_{0}[3] &= \omega^{\varepsilon_{0}[2]} \\ &= \omega^{\omega^{\varepsilon_{0}[1]}} \\ &= \omega^{\omega^{\omega^{\varepsilon_{0}[0]}}} \\ &= \omega^{\omega^{\omega^{0}[3]}} \\ &= \omega^{\omega^{1}[3]} \\ &= \omega^{\omega[3]} \\ &= \omega^{3}[3] \\ &= \omega^{2}\cdot\omega[3] \\ &= \omega^{2}\cdot3[3] \\ &= \omega^{2}\cdot2+\omega^{2}[3] \\ &= \omega^{2}\cdot2+\omega\cdot\omega[3] \\ &= \omega^{2}\cdot2+\omega\cdot3[3] \\ &= \omega^{2}\cdot2+\omega\cdot2+\omega[3] \\ &\phantom{=} \vdots \quad \raisebox{0.2em}{\text{after 3 more steps}} \\ &= \omega^{2}\cdot2+\omega\cdot2+3 \\ \end{align*} $$
$$ \begin{align*} f_{\omega^{\omega^{\omega}}}(2) &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+2}(2) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(2)) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(2))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+2}(2))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(2)))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega^{\omega}}(2))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+2}(2))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega+1}(2)))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega}(f_{\omega^{\omega+1}+\omega}(2))))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega}(f_{\omega^{\omega+1}+2}(2))))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega}(f_{\omega^{\omega+1}+1}(f_{\omega^{\omega+1}+1}(2)))))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega}(f_{\omega^{\omega+1}+1}(f_{\omega^{\omega+1}}(f_{\omega^{\omega+1}}(2))))))))) \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega}(f_{\omega^{\omega+1}+1}(f_{\omega^{\omega+1}}(f_{\omega^{\omega}+\omega+2}(2))))))))) \\ &\phantom{=} \vdots \quad \raisebox{0.2em}{\text{after 56 more steps}} \\ &= f_{\omega^{\omega+1}+\omega^{\omega}+\omega+1}(f_{\omega^{\omega+1}+\omega^{\omega}+\omega}(f_{\omega^{\omega+1}+\omega^{\omega}+1}(f_{\omega^{\omega+1}+\omega^{\omega}}(f_{\omega^{\omega+1}+\omega+1}(f_{\omega^{\omega+1}+\omega}(f_{\omega^{\omega+1}+1}(f_{\omega^{\omega+1}}(f_{\omega^{\omega}+\omega+1}(f_{\omega^{\omega}+\omega}(f_{\omega^{\omega}+1}(f_{\omega^{\omega}}(f_{\omega+1}(f_{8}(8)))))))))))))) = \dots\\ \end{align*} $$
$$ \begin{align*} f_{\omega^{\omega^{\omega}}}(3) &= f_{\omega^{\omega^{2}\cdot2+\omega\cdot2+2}\cdot2+\omega^{\omega^{2}\cdot2+\omega\cdot2+1}\cdot2+\omega^{\omega^{2}\cdot2+\omega\cdot2}\cdot2+\omega^{\omega^{2}\cdot2+\omega+2}\cdot2+\omega^{\omega^{2}\cdot2+\omega+1}\cdot2+\omega^{\omega^{2}\cdot2+\omega}\cdot2+\omega^{\omega^{2}\cdot2+2}\cdot2+\omega^{\omega^{2}\cdot2+1}\cdot2+\omega^{\omega^{2}\cdot2}\cdot2+\omega^{\omega^{2}+\omega\cdot2+2}\cdot2+\omega^{\omega^{2}+\omega\cdot2+1}\cdot2+\omega^{\omega^{2}+\omega\cdot2}\cdot2+\omega^{\omega^{2}+\omega+2}\cdot2+\omega^{\omega^{2}+\omega+1}\cdot2+\omega^{\omega^{2}+\omega}\cdot2+\omega^{\omega^{2}+2}\cdot2+\omega^{\omega^{2}+1}\cdot2+\omega^{\omega^{2}}\cdot2+\omega^{\omega\cdot2+2}\cdot2+\omega^{\omega\cdot2+1}\cdot2+\omega^{\omega\cdot2}\cdot2+\omega^{\omega+2}\cdot2+\omega^{\omega+1}\cdot\omega[3]}(3) = \dots\\ \end{align*} $$
$$ \varepsilon_{0}\cdot\omega[3]=\varepsilon_{0}\cdot2+\omega^{2}\cdot2+\omega\cdot2+3 $$
$$ \varphi(1, 1)[0]=\varepsilon_{0}+1 $$
$$ \begin{align*} \varphi(1, 1)[1] &= \omega^{\varphi(1, 1)[0]} \\ &= \omega^{\varepsilon_{0}+1}[1] \\ &= \omega^{\varepsilon_{0}+1}[1] = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(1, 1)[2] &= \omega^{\varphi(1, 1)[1]} \\ &= \omega^{\omega^{\varphi(1, 1)[0]}} \\ &= \omega^{\omega^{\varepsilon_{0}+1}[2]} \\ &= \omega^{\omega^{\varepsilon_{0}+1}[2]} = \dots\\ \end{align*} $$
$$ f_{\omega^{\omega^{2}\cdot2}}(3)=f_{\omega^{\omega^{2}+\omega\cdot2+2}\cdot2+\omega^{\omega^{2}+\omega\cdot2+1}\cdot2+\omega^{\omega^{2}+\omega\cdot2}\cdot2+\omega^{\omega^{2}+\omega+2}\cdot2+\omega^{\omega^{2}+\omega+1}\cdot2+\omega^{\omega^{2}+\omega}\cdot2+\omega^{\omega^{2}+2}\cdot2+\omega^{\omega^{2}+1}\cdot2+\omega^{\omega^{2}}\cdot2+\omega^{\omega\cdot2+2}\cdot2+\omega^{\omega\cdot2+1}\cdot2+\omega^{\omega\cdot2}\cdot2+\omega^{\omega+2}\cdot2+\omega^{\omega+1}\cdot2+\omega^{\omega}\cdot2+\omega^{2}\cdot2+\omega\cdot2+3}(3)=\dots $$
$$ \begin{align*} \varphi(1, \varphi(1))[2] &= \varepsilon_{\varphi(1)[2]} \\ &= \varepsilon_{\omega^{1}[2]} \\ &= \varepsilon_{\omega[2]} \\ &= \varepsilon_{2}[2] \\ &= \omega^{\varepsilon_{2}[1]} \\ &= \omega^{\omega^{\varepsilon_{2}[0]}} \\ &= \omega^{\omega^{\varepsilon_{1}+1}[2]} \\ &= \omega^{\omega^{\varepsilon_{1}+1}[2]} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(2, 0)[3] &= \varepsilon_{\varphi(2, 0)[2]} \\ &= \varepsilon_{\varepsilon_{\varphi(2, 0)[1]}} \\ &= \varepsilon_{\varepsilon_{\varepsilon_{\varphi(2, 0)[0]}}} \\ &= \varepsilon_{\varepsilon_{\varepsilon_{0}[3]}} \\ &= \varepsilon_{\varepsilon_{\omega^{\varepsilon_{0}[2]}}} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(2, \omega)[2] &= \xi_{\omega[2]} \\ &= \xi_{2}[2] \\ &= \varepsilon_{\xi_{2}[1]} \\ &= \varepsilon_{\varepsilon_{\xi_{2}[0]}} \\ &= \varepsilon_{\varepsilon_{\xi_{1}+1}[2]} \\ &= \varepsilon_{\omega^{\varepsilon_{\xi_{1}+1}[1]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{1}+1}[0]}}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{1}}+1}[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{1}}+1}[2]}} = \dots\\ \end{align*} $$
$$ \begin{align*} f_{\varphi(4, 1)}(3) &= f_{\eta_{\eta_{\xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\omega^{\omega^{\varepsilon_{\xi_{\eta_{\varphi(4, 0)}}}}\cdot2+\omega^{\color{black}{\varepsilon_{\xi_{\eta_{\eta_{\eta_{\xi_{\xi_{\varepsilon_{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+\omega\cdot2+2}}\cdot2+\omega^{\color{blue}{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+\omega\cdot2+1}}\cdot2+\omega^{\color{green}{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+\omega\cdot2}}\cdot2+\omega^{\color{orange}{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+\omega+2}}\cdot2+\omega^{\color{brown}{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+\omega+1}}\cdot2+\omega^{\color{purple}{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+\omega}}\cdot2+\omega^{\color{red}{\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+2}}\cdot2+\omega^{\omega^{\omega^{\varepsilon_{\omega^{2}\cdot2+2}[1]}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}(3) = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\omega, 0)[2] &= \varphi(\omega[2], 0) \\ &= \xi_{0}[2] \\ &= \varepsilon_{\xi_{0}[1]} \\ &= \varepsilon_{\varepsilon_{\xi_{0}[0]}} \\ &= \varepsilon_{\varepsilon_{0}[2]} \\ &= \varepsilon_{\omega^{\varepsilon_{0}[1]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{0}[0]}}} \\ &= \varepsilon_{\omega^{\omega^{0}[2]}} \\ &= \varepsilon_{\omega^{1}[2]} \\ &= \varepsilon_{\omega[2]} \\ &= \varepsilon_{2}[2] = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\omega, 0)[3] &= \varphi(\omega[3], 0) \\ &= \eta_{0}[3] \\ &= \xi_{\eta_{0}[2]} \\ &= \xi_{\xi_{\eta_{0}[1]}} \\ &= \xi_{\xi_{\xi_{\eta_{0}[0]}}} \\ &= \xi_{\xi_{\xi_{0}[3]}} \\ &= \xi_{\xi_{\varepsilon_{\xi_{0}[2]}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\xi_{0}[1]}}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\varepsilon_{\xi_{0}[0]}}}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\varepsilon_{0}[3]}}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\varepsilon_{0}[2]}}}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\omega^{\varepsilon_{0}[1]}}}}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\omega^{\omega^{\varepsilon_{0}[0]}}}}}}} \\ &\phantom{=} \vdots \quad \raisebox{0.2em}{\text{after 9 more steps}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\omega[3]}}}}} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\omega, 1)[2] &= \xi_{\varphi(\omega, 0)+1}[2] \\ &= \varepsilon_{\xi_{\varphi(\omega, 0)+1}[1]} \\ &= \varepsilon_{\varepsilon_{\xi_{\varphi(\omega, 0)+1}[0]}} \\ &= \varepsilon_{\varepsilon_{\xi_{\varphi(\omega, 0)}+1}[2]} \\ &= \varepsilon_{\omega^{\varepsilon_{\xi_{\varphi(\omega, 0)}+1}[1]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 0)}+1}[0]}}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 0)}}+1}[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 0)}}}\cdot\omega[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 0)}}}\cdot\omega[2]}} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\omega, \omega)[2] &= \varphi(\omega, \omega[2]) \\ &= \varphi(\omega, 2)[2] \\ &= \xi_{\varphi(\omega, 1)+1}[2] \\ &= \varepsilon_{\xi_{\varphi(\omega, 1)+1}[1]} \\ &= \varepsilon_{\varepsilon_{\xi_{\varphi(\omega, 1)+1}[0]}} \\ &= \varepsilon_{\varepsilon_{\xi_{\varphi(\omega, 1)}+1}[2]} \\ &= \varepsilon_{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}+1}[1]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}+1}[0]}}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}+1}[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}}\cdot\omega[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}}\cdot2[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}}+\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}}[2]}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}}+\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}[2]}}} \\ &\phantom{=} \vdots \quad \raisebox{0.2em}{\text{after 3 more steps}} \\ &= \varepsilon_{\omega^{\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)}}}+\omega^{\varepsilon_{\xi_{\varphi(\omega, 1)[2]}}}}} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\omega, 2)[3] &= \eta_{\varphi(\omega, 1)+1}[3] \\ &= \xi_{\eta_{\varphi(\omega, 1)+1}[2]} \\ &= \xi_{\xi_{\eta_{\varphi(\omega, 1)+1}[1]}} \\ &= \xi_{\xi_{\xi_{\eta_{\varphi(\omega, 1)+1}[0]}}} \\ &= \xi_{\xi_{\xi_{\eta_{\varphi(\omega, 1)}+1}[3]}} \\ &= \xi_{\xi_{\varepsilon_{\xi_{\eta_{\varphi(\omega, 1)}+1}[2]}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\xi_{\eta_{\varphi(\omega, 1)}+1}[1]}}}} \\ &= \xi_{\xi_{\varepsilon_{\varepsilon_{\varepsilon_{\xi_{\eta_{\varphi(\omega, 1)}+1}[0]}}}}} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\varphi(\varphi(\omega, 0), 0), 0)[3] &= \varphi(\varphi(\varphi(\omega, 0), 0)[3], 0) \\ &= \varphi(\varphi(\varphi(\omega, 0)[3], 0), 0) \\ &= \varphi(\varphi(\varphi(\omega[3], 0), 0), 0) \\ &= \varphi(\varphi(\eta_{0}[3], 0), 0) \\ &= \varphi(\varphi(\xi_{\eta_{0}[2]}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\eta_{0}[1]}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\xi_{\eta_{0}[0]}}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\xi_{0}[3]}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\varepsilon_{\xi_{0}[2]}}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\varepsilon_{\varepsilon_{\xi_{0}[1]}}}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\varepsilon_{\varepsilon_{\varepsilon_{\xi_{0}[0]}}}}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\varepsilon_{\varepsilon_{\varepsilon_{0}[3]}}}}, 0), 0) \\ &= \varphi(\varphi(\xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\varepsilon_{0}[2]}}}}}, 0), 0) \\ &\phantom{=} \vdots \quad \raisebox{0.2em}{\text{after 13 more steps}} \\ &= \varphi(\varphi(\xi_{\xi_{\varepsilon_{\varepsilon_{\omega^{\omega[3]}}}}}, 0), 0) = \dots\\ \end{align*} $$
$$ \varphi(3)[2]=\omega^{3}[2]=\dots $$
$$ \varphi(4, 0, 0)[0]=0 $$
$$ \begin{align*} \Gamma_{0}[3] &= \varphi(\Gamma_{0}[2], 0) \\ &= \varphi(\varphi(\Gamma_{0}[1], 0), 0) \\ &= \varphi(\varphi(\varphi(\Gamma_{0}[0], 0), 0), 0) \\ &= \varphi(\varphi(\omega^{0}[3], 0), 0) \\ &= \varphi(\varepsilon_{0}[3], 0) \\ &= \varphi(\omega^{\varepsilon_{0}[2]}, 0) \\ &= \varphi(\omega^{\omega^{\varepsilon_{0}[1]}}, 0) \\ &= \varphi(\omega^{\omega^{\omega^{\varepsilon_{0}[0]}}}, 0) \\ &= \varphi(\omega^{\omega^{\omega^{0}[3]}}, 0) \\ &= \varphi(\omega^{\omega^{1}[3]}, 0) \\ &= \varphi(\omega^{\omega[3]}, 0) \\ &= \varphi(\omega^{3}[3], 0) \\ &= \varphi(\omega^{2}\cdot\omega[3], 0) \\ &\phantom{=} \vdots \quad \raisebox{0.2em}{\text{after 27 more steps}} \\ &= \varphi(\omega^{2}\cdot2+\omega\cdot2+2, \varphi(\omega^{2}\cdot2+\omega\cdot2+2, \varphi(\omega^{2}\cdot2+\omega\cdot2+2, 0)[3])) = \dots\\ \end{align*} $$
$$ \begin{align*} \Gamma_{1}[3] &= \varphi(\Gamma_{1}[2], 0) \\ &= \varphi(\varphi(\Gamma_{1}[1], 0), 0) \\ &= \varphi(\varphi(\varphi(\Gamma_{1}[0], 0), 0), 0) \\ &= \varphi(\varphi(\varphi(\Gamma_{0}+1, 0)[3], 0), 0) \\ &= \varphi(\varphi(\varphi(\Gamma_{0}, \varphi(\Gamma_{0}+1, 0)[2]), 0), 0) = \dots\\ \end{align*} $$
$$ \begin{align*} \Gamma_{\omega}[3] &= \Gamma_{\omega[3]} \\ &= \Gamma_{3}[3] \\ &= \varphi(\Gamma_{3}[2], 0) = \dots\\ \end{align*} $$
$$ \begin{align*} \Gamma_{\Gamma_{0}}[3] &= \Gamma_{\Gamma_{0}[3]} \\ &= \Gamma_{\varphi(\Gamma_{0}[2], 0)} \\ &= \Gamma_{\varphi(\varphi(\Gamma_{0}[1], 0), 0)} \\ &= \Gamma_{\varphi(\varphi(\varphi(\Gamma_{0}[0], 0), 0), 0)} \\ &= \Gamma_{\varphi(\varphi(\omega^{0}[3], 0), 0)} \\ &= \Gamma_{\varphi(\varepsilon_{0}[3], 0)} \\ &= \Gamma_{\varphi(\omega^{\varepsilon_{0}[2]}, 0)} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(1, 1, 0)[3] &= \Gamma_{\varphi(1, 1, 0)[2]} \\ &= \Gamma_{\Gamma_{\varphi(1, 1, 0)[1]}} \\ &= \Gamma_{\Gamma_{\Gamma_{\varphi(1, 1, 0)[0]}}} \\ &= \Gamma_{\Gamma_{\Gamma_{0}[3]}} \\ &= \Gamma_{\Gamma_{\varphi(\Gamma_{0}[2], 0)}} = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(1, \omega, 0)[3] &= \varphi(1, \omega[3], 0) \\ &= \varphi(1, 3, 0)[3] \\ &= \varphi(1, 2, \varphi(1, 3, 0)[2]) \\ &= \varphi(1, 2, \varphi(1, 2, \varphi(1, 3, 0)[1])) \\ &= \varphi(1, 2, \varphi(1, 2, \varphi(1, 2, \varphi(1, 3, 0)[0]))) \\ &= \varphi(1, 2, \varphi(1, 2, \varphi(1, 2, 0)[3])) \\ &= \varphi(1, 2, \varphi(1, 2, \varphi(1, 1, \varphi(1, 2, 0)[2]))) \\ &= \varphi(1, 2, \varphi(1, 2, \varphi(1, 1, \varphi(1, 1, \varphi(1, 2, 0)[1])))) = \dots\\ \end{align*} $$
$$ \varphi(1, \Gamma_{1}, 0)[3]=\varphi(1, \varphi(\varphi(\varphi(\Gamma_{0}, \varphi(\Gamma_{0}+1, 0)[2]), 0), 0), 0)=\dots $$
$$ \varphi(1, \varphi(1, 1, 0), 0)[3]=\varphi(1, \Gamma_{\Gamma_{\varphi(\Gamma_{0}[2], 0)}}, 0)=\dots $$
$$ \varphi(\omega, 0, 0)[3]=\varphi(2, \varphi(2, \varphi(1, \varphi(1, \varphi(\varphi(\Gamma_{0}[1], 0), 0), 0), 0), 0), 0)=\dots $$
$$ \varphi(\omega+1, 0, 0)[3]=\varphi(\omega, \varphi(\omega, \varphi(2, \varphi(2, \varphi(1, \varphi(1, \Gamma_{0}[3], 0), 0), 0), 0), 0), 0)=\dots $$
$$ \begin{align*} \varphi(\omega, 0, 1)[3] &= \varphi(3, \varphi(\omega, 0, 0)+1, 0)[3] = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(\varepsilon_{0}, \omega, 0, \omega)[3] &= \varphi(\varepsilon_{0}, \omega, 0, \omega[3]) \\ &= \varphi(\varepsilon_{0}, \omega, 0, 3)[3] \\ &= \varphi(\varepsilon_{0}, 3, \varphi(\varepsilon_{0}, \omega, 0, 2)+1, 0)[3] \\ &= \varphi(\varepsilon_{0}, 3, \varphi(\varepsilon_{0}, \omega, 0, 2), \varphi(\varepsilon_{0}, 3, \varphi(\varepsilon_{0}, \omega, 0, 2)+1, 0)[2]) = \dots\\ \end{align*} $$
$$ \begin{align*} \varphi(1, 0, 0, 0, 0)[3] &= \varphi(\varphi(1, 0, 0, 0, 0)[2], 0, 0, 0) \\ &= \varphi(\varphi(\varphi(1, 0, 0, 0, 0)[1], 0, 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\varphi(\varphi(1, 0, 0, 0, 0)[0], 0, 0, 0), 0, 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\omega^{0}[3], 0, 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(1, 0, 0, 0)[3], 0, 0, 0) \\ &= \varphi(\varphi(\varphi(1, 0, 0, 0)[2], 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\varphi(\varphi(1, 0, 0, 0)[1], 0, 0), 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\varphi(\varphi(\varphi(1, 0, 0, 0)[0], 0, 0), 0, 0), 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\varphi(\omega^{0}[3], 0, 0), 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\Gamma_{0}[3], 0, 0), 0, 0, 0) \\ &= \varphi(\varphi(\varphi(\Gamma_{0}[2], 0), 0, 0), 0, 0, 0) = \dots\\ \end{align*} $$
$$ \varphi(\omega\mathbin{\char64}0)[3]=\omega^{\omega}[3]=\dots $$
$$ \varphi(1\mathbin{\char64}\omega)[3]=\varphi(1\mathbin{\char64}3)[3]=\dots $$
$$ \begin{align*} \varphi(2\mathbin{\char64}\omega)[3] &= {\begin{pmatrix} 1 & 1 \\ \omega & \omega[3] \end{pmatrix}} \\ &= {\begin{pmatrix} 1 & 1 \\ \omega & 3 \end{pmatrix}}[3] \\ &= {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & 1 \\ \omega & 3 \end{pmatrix}}[2] \\ \omega & 2 \end{pmatrix}} \\ &= {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & 1 \\ \omega & 3 \end{pmatrix}}[1] \\ \omega & 2 \end{pmatrix}} \\ \omega & 2 \end{pmatrix}} \\ &= {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & 1 \\ \omega & 3 \end{pmatrix}}[0] \\ \omega & 2 \end{pmatrix}} \\ \omega & 2 \end{pmatrix}} \\ \omega & 2 \end{pmatrix}} \\ &= {\begin{pmatrix} 1 & {\begin{pmatrix} 1 & \varphi(1\mathbin{\char64}\omega)[3] \\ \omega & 2 \end{pmatrix}} \\ \omega & 2 \end{pmatrix}} = \dots\\ \end{align*} $$
$$ \begin{align*} {\begin{pmatrix} 1 & 1 \\ \omega & 0 \end{pmatrix}}[3] &= \varphi(\varphi(1\mathbin{\char64}\omega)+1\mathbin{\char64}\omega[3]) \\ &= \varphi(\varphi(1\mathbin{\char64}\omega)+1\mathbin{\char64}3)[3] \\ &= {\begin{pmatrix} \varphi(1\mathbin{\char64}\omega) & \varphi(\varphi(1\mathbin{\char64}\omega)+1\mathbin{\char64}3)[2] \\ 3 & 2 \end{pmatrix}} = \dots\\ \end{align*} $$
$$ \begin{align*} {\begin{pmatrix} 2 & \omega \\ \omega & 0 \end{pmatrix}}[3] &= {\begin{pmatrix} 2 & \omega[3] \\ \omega & 0 \end{pmatrix}} \\ &= {\begin{pmatrix} 2 & 3 \\ \omega & 0 \end{pmatrix}}[3] \\ &= {\begin{pmatrix} 1 & {\begin{pmatrix} 2 & 2 \\ \omega & 0 \end{pmatrix}}+1 \\ \omega & \omega[3] \end{pmatrix}} \\ &= {\begin{pmatrix} 1 & {\begin{pmatrix} 2 & 2 \\ \omega & 0 \end{pmatrix}}+1 \\ \omega & 3 \end{pmatrix}}[3] = \dots\\ \end{align*} $$
$$ \varphi(\varepsilon_{0}\mathbin{\char64}\varepsilon_{0})[3]={\begin{pmatrix} \omega^{2}\cdot2+\omega\cdot2+2 & 1 \\ \varepsilon_{0} & \varepsilon_{0}[3] \end{pmatrix}}=\dots $$
$$ \begin{align*} {\begin{pmatrix} \omega & 1 \\ \omega & 0 \end{pmatrix}}[3] &= {\begin{pmatrix} \omega[3] & \varphi(\omega\mathbin{\char64}\omega)+1 \\ \omega & \omega[3] \end{pmatrix}} \\ &= {\begin{pmatrix} \omega[3] & \varphi(\omega\mathbin{\char64}\omega)+1 \\ \omega & 3 \end{pmatrix}}[3] \\ &= {\begin{pmatrix} \omega[3] & \varphi(\omega\mathbin{\char64}\omega) & {\begin{pmatrix} \omega[3] & \varphi(\omega\mathbin{\char64}\omega)+1 \\ \omega & 3 \end{pmatrix}}[2] \\ \omega & 3 & 2 \end{pmatrix}} = \dots\\ \end{align*} $$
$$ \varphi(1\mathbin{\char64}\varphi(1\mathbin{\char64}\omega))[3]=\varphi(1\mathbin{\char64}\varphi({\begin{pmatrix} \omega^{2}\cdot2+\omega\cdot2+2 & \varphi(\omega^{2}\cdot2+\omega\cdot2+3\mathbin{\char64}1)[2] \\ 1 & 0 \end{pmatrix}}\mathbin{\char64}2))=\dots $$
$$ \dots \text{more cases to come} \dots $$
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