User blog comment:B1mb0w/FGH of Omega/@comment-27513631-20160608172607

Why should I believe that \(f_\omega(\omega)>\varphi(\omega,0)\)?

\(\f_0(\alpha) = \alpha+1\)

\(\f_1(\alpha) = \alpha\cdot 2\)

\(\f_2(\alpha) = \alpha\cdot2^\alpha < \omega^{\alpha\cdot 2}\)

\(\f_3(\alpha) = f_2^\alpha(\alpha) \leq \varepsilon_\alpha\)

\(\f_4(\alpha) = f_3^\alpha(\alpha) \leq \zeta_\alpha\)

\(\f_{n+3}(\alpha) \leq \varphi(n+1,\alpha)\)

and, more generally,

\(\f_{\beta}(\alpha) \leq \varphi(\beta,\alpha)\)