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

"it is also quite clear that we do not even have to assign a fundamental sequence..."

This is not true. It is possible to construct pathological fundamental sequences to force unwanted behavior. I'll even give an example:

Look at the Goodstein sequences, and their lengths G(n). Using our "usual" fundamental sequences, G(n) has the growth rate of approximately \(f_{\varepsilon_0}\). So let's define a new system of fundamental sequences. In our new system, all fundamental sequences are the same, except for that of \(\omega\). We define \(\omega [n] = G(n)\). Then clearly we have that \(f_{\varepsilon_0}\) grows slower than \(f_\omega\)!