User blog comment:D57799/feeding FGH into FGH/@comment-5529393-20141005173539

Interesting post. However, there's no reason why the catchup ordinal for when G_a(n) catches up with F_a(n) should be equal to the ordinal when G_a(w) is approximately F_a(w). If we assume that G_a(w) = a, then the definition I gave for F_a(w), which seems natural, has them meeting up at Gamma_0. So we would need a much more powerful construction of F_a(w); come up with that, and you've got something.

Incidentally, it's not that there are arguments about when the first catchup point occurs, it's that it depends on the choice of fundamental sequences. For the first choice of fundamental sequences for which this question was studied, the catchup ordinal was proven to be $$\psi_0(\Omega_\omega)$$, but Andreas Weiermann proved later that, for various "natural" choices of fundamental sequences, we could achieve vastly different catchup ordinals. (like $$\varepsilon_0$$ or $$\omega^2$$)