User blog comment:SuperJedi224/FGH with ordinals/@comment-27173506-20151217143857

If you fill in logical gaps, it's quite easy to see that from f_3(w) onwards it just describes the Veblen hierarchy, as each f_n+1(w) it just describes f_n(f_n(f_n... infinite times, exactly like the (as I said) Veblen hierarchy. In general, for n > 0, f_n+2(w) = phi(n,0). Therefore, f_w(w) ~ phi(w,0).

We can extrapolate from here, and see that f_w+1(w) is approximately phi(1,0,0), as it is the limit of the series: {f_w(w) ~ phi(w,0), f_w(f_w(w)) ~ phi(phi(w,0),0), ...}. Without much difficulty you can show that f_w+2(w) is approximately phi(1,0,0,0) and f_w2(w) is about SVO.

From here it gets a bit more complicated, but maybe I'll do it one day.