User blog:Allam948736/Some ordinal questions

In one possible interpretation of ordinal hyperoperators, \(\omega \uparrow\uparrow \omega+1 = \varphi(\omega, 0)\) instead of just \(\varepsilon_1\) or \(\varepsilon_{\omega^\omega}\). I have analyzed this "3rd interpretation" further and found that \(\omega \uparrow\uparrow \omega^2 = \Gamma_0\), \(\omega \uparrow\uparrow \omega^\omega = \vartheta(\Omega^\omega)\), and that \(\omega \uparrow\uparrow\uparrow \omega = \vartheta(\Omega^\Omega)\). However, I don't know enough about ordinals at that level to continue further. What would (\omega \uparrow\uparrow\uparrow\uparrow \omega\) be? \(\omega \uparrow^{\omega \uparrow^{\omega \uparrow...}}\)?

What is the ordinal \(\varphi(\Gamma_0, 1)\) equal to? Is it the limit of \(\varphi(1, \Gamma_0 + 1)\), \(\varphi(\varphi(1, 0), \Gamma_0 + 1)\), ...?