User blog comment:LittlePeng9/First order oodle theory/@comment-10429372-20141030174048

Can we continue to extend by making a language that actually can use different types of brackets, so that it can define \(\xi(\Omega)\) and much higher \(\xi\) functions, further it can define \(\xi^{CK}_1\), \(\xi^{FOOT}_1\) (yep I just made this up, \(\omega^{FOOT}_1\) is the growth rate of FOOT(n)). Call the definded function \(\xi^{FOOT}_1\)-FOOT(n), than \(f_{\omega^{FOOT}_2}(n)\) is comparable to it. Now extend up to the omega-FOOT fixed point, which is also the xi-FOOT fixed point.

But using \(\xi_1\) we can go further. We can even define OFOOT(n), and when we get to O'FOOT(n), we just have SOOT(n). Now extend to O'OOT(n), being FOOOT(n), well... you guess where this will lead. FOwT(n), FOOT(n), and that's about the limit I guess.