User blog comment:Edwin Shade/Rank-on-Rank Turing Ordinals and Beyond/@comment-26454151-20180120005845/@comment-1605058-20180120095827

No, if we define \(\omega[n]=G(n)\), it's not true that \(f_{\varepsilon_0}\) grows slower than \(f_\omega\) - what's true is that \(f_{\varepsilon_0}\) using the usual fundamental sequences is growing slower than \(f_\omega\) using the new fundamental sequences.

Similarly, if we define \(\omega[n]=\Sigma(n)\) we will still have \(f_{\omega_1^{CK}}>^*f_\omega\), but the difference is that \(f_\omega\) is an uncomputable function growing a bit faster than \(\Sigma(n)\), and \(f_{\omega_1^{CK}}\) is an even faster growing uncomputable function.